We have books full of evidence: http://www.theflatearthsociety.org/home/index.php/library
If you are more of a poetry guy or want the quick version, I'd suggest looking at Lady Blount's the Why and Because: http://www.theflatearthsociety.org/home/index.php/featured/the-why-and-because
Then I will provide a simple answer. The evidence is the same evidence that leads one to believe the Earth is Round. We aren't viewing some special set of data the rest of the world is missing. We are just interpreting it, we believe, correctly.
There's a lot of indirect evidence that convinces folks. Some is valid, some is not - and yet that is the evidence that leads them to believe what they do.
I've had interested new flatists message me and talk about why they came to believe. They can be divided into three broad categories: religious, everyday experience, and conspiracy. Now and then you'll get the philosopher or epistemologist that has their own reasons. I will supply a sampling of these and leave it to you to read the texts aforelinked. There is also epistemological and philosophical basis for our views that lend evidence to the point which I can talk to if you'd like.
A few everyday evidences:
1) Find the curve! You can't. Every experience we have says it's flat - from the beach to Mount Everest to a plane.
2) Water has been determined experimentally to be level. If the earth was round, and spun, water would not be level.
3) No consideration is made in architecture of large constructions for curvature of earth.
4) No consider is made in the construction of the Suez Canal - it forms a straight and level surface of water (Carpenter)
5) Rivers such as the Nile would have to flow uphill
6) Lighthouses can be seen further than one would expect. Cape Hatteras is seen at 40 miles, which should be 900 feet above the sea level to be visible. This is not the case (Carpenter)
7) The horizon rises to eye level
We'd feel the rotational acceleration.
9) So would projectiles - we've talked to experts and they say the don't account for that. Snipers and Ballistics experts we have interviewed do not account for coriolis effects (Mark Sargent)
10) Train tracks degrade eveny on both sides - if the earth rotated they would be more worn on one side. Railroads have even wear in spite of Coriolis effects.
11) Air does not feel this acceleration. Winds aren't rushing by us in one direction as gravity would dictate.
12) Everything we know and feel says it doesn't spin.
13) How are the moon phases are lit in seemingly impossible ways when both the sun and moon are in sky?
14) Daytime lunar eclipses as noted by Kepler
15) Antarctic extended daylight times did not match up for me with expected values from almanacs and on board navigation experts
16) Navigation at sea is done through "Plane Sailing"
17) No difference in the distance bullets travel against or with the supposed rotation (Carpenter)
18) Foucault Pendulums must be controlled magnetically to achieve real results as they are in most museums
19) The Bedford Level Experiment
20) Lack of Curvature in Planes
Conspiracy Evidences:
21) Lack of ability to provide us with a picture of the globe unaltered and spinning
22) Constant lies and photo manipulation from Camp NASA
23) General Byrds Testimony
24) Antarctic Treaties
25) Instead of swearing on the Bible the went to space, astronauts punch those who ask
26) Leaving the moon would be impossible
27) Stars don't exist
A few religious evidences:
28) Facing Mecca would be impossible
29) Genesis describes a flat earth
30) Jewish Cosmology describes a dome model
31) Buddhism and Mount meru describe a flat earth
32) Shinto Describes a flat earth
33) Native American Religions Describe a flat earth
34) Taoists discovered a flat earth
35) Every religion holds the earth to be flat.
36) Jesus viewing all the kingdoms of the world
and so on. I could literally list thousands.
firstly, i wouldn't describe these as 'currently accepted proofs' as the majority have already been disproven, and even accepted by a huge majority of flat earthers, I'll go through them one by one though, as you have
1. curve - there IS a curve, most flat earth videos trying to disprove a curve, at some point, will have half of a landmark showing.. and unless there's some half above water and half below water cities that i'm not aware of, then that's absolute proof of a curve, how anyone is expecting to see a curve with the sheer size of the planet anyway is beyond me!
also, when there's photos of landmarks from too far away, the camera is never at ground level, making the tests void, when the camera is put at ground level, the difference you can see is NOT around 6ft, its a drastic difference, there's good reason lighthouses are built on the top of cliffs etc
2. water level - well this is gravity related, i'm not scientific enough to get into a detailed gravity debate.. the way i understand it is; its like a magnet in the centre of the earth, everything within the atmosphere is attracted to it (and only huge bodies outside of the atmosphere are attracted to it, the bigger the mass etc)
so, with my BASIC understanding of gravity, i've no reason to see the water levelling out in accordance with the magnetic cente outwards, creating a shperical shape, as any kind of issue whatsoever, but again, i don't understand the ins and outs so i'm not claiming anything

3. buildings - is this serious? how big do people think buildings are? nowhere NEAR large enough to have any kind of need to take the curvature into account, if anyone uses this as any kind of proof, they need to go right back to the basics
4. suez canal - I can't really comment as i've not really looked into it much, a quick search shows people with photos of the top half of ships on the suez though, photos though, urgh right? lol
5. the nile - uphill? why? it flows from south to north, but it does happen to be in a downhill direction.. do ANY of these flat earthers you're quoting know how to cross examine?
6. lighthouses - see answer 1, or answer the question; why are lighthouses so often built so high above sea level
7. horizon rising to eye level - thats basic perspective, you're too far away from the horizon to be able to tell exactly where your eye level is above the horizon you're looking at, it would be impossible to look at the point that is exactly your own height above the horizon
8. feeling the rotation - i'm not sure why someone would expect to feel any rotation spin, its the same as being in a train and being on top of a train, you don't feel the wind in the train as you're encased IN the train.. you don't feel the wind in the atmosphere as you're encased IN the atmosphere, gravity is affecting the whole atmosphere, it doesn't stop at the planet, otherwise if you threw a ball up, it wouldn't come back down, again, very basic stuff that even I can explain
9. corielis effect - i'm not sure who you've spoken to, but after the amount of flat earthers screaming 'don't just believe what you're told', then i'll do the same.. do the tests yourselves, i've seen them done (on video)
this is where there's a major stumbling block, until a flat earther actually does any physical testing themselves, they just cry 'fake' to every test, every first hand account, every video.. so there's no way of proving anything without convincing a flat earther to test things themselves, which most are unwilling to do, the same goes for the 'landmarks from too far away' test, just DO the test, they will then have irrefutable evidence..
..but they won't
the corielis effect also affects storm rotations and water rotations (toilets, sinks, bathtubs, whirlpools etc)
i've heard the answer for the storms, fair enough, it works on both models to an extent
but the water answers are usually pretty few and far between, when they do come they're flaky as hell, mainly 'they're made that way'.. well i'm damn sure if I imported a sink from Australia it aint gonna spin the opposite direction to my french imported bathtub so that's that not a valid answer.. also, the corielis effect isn't necessary due to stronger proof, i normally just agree to disagee, the effect is miniscule anyway
10. train tracks - see answer 8, people actually ask this? i've never once heard this, secondly, it would ONLY be relevant if train tracks only ran in straight lines, from (what us globe earthers would call) south to north / north to south
11. i'm assuming you meant;
Air does not feel this acceleration. Winds aren't rushing by us in one direction as rotation would dictate...?
if so, see answer 8
if not, i'll assume you mean that you're expecting gravity to pull air downwards towards the ground?
gravity needs mass, there's no mass with air?
again, gravity is definitely NOT my strongpoint
12.
Everything we know and feel says it doesn't spin again, see answer 8.. with the KNOWLEDGE we do have, we understand why we don't FEEL the spin, the train analogy is a perfect example
13. crescent moon during the day - another what can only be described as a shockingly basic question; remember, the moonrise and moonset have nothing to do with the sunrise and sunset, the moon has its own schedule.. i'm not even sure i understand the point that's supposed to be making
14. daytime lunar eclipse - i've answered this on here many times.. when this has been reported, both the sun and the moon were at opposite ends of the horizon.. this is due to 'refraction'
this is usually met, strangely enough, with flat earthers claiming that its impossible for that to be correct (i've not had any elaboration on a standard 'you're wrong', so i'm not sure of their reasoning)
..to which I tell them they then have to RE-answer their 'refraction' claim for the sun going DOWN below the horizon instead of disappearing to a point as you would expect if it was going away
on one hand i'm told it cannot cause that effect, on the other i'm told its evidence
15. i cannot really comment on that, your tests are your tests, i know all i need to know about Antarctica, its of very little actual significance to the flat earth theory to be honest, its of huge significance to the anti elite, but not the flat earth theory.. i'll elaborate in another answer
16. plane sailing - as far as i can understand, plain sailing is used to find 'approximate' distances instead of having to take any curvature into account using angular mathematics etc, to simply make things easier, hence the derived expression
17. see answer 9, already covered the corielis effect
18. pendulums - i feel things like this are just clutching at straws, pendulums don't naturally go on forever, of course you're going to need magnetism for a 24hr test, the magnets are turned on and off automatically too
19. bedford level experiment - i'm not sure what you're saying about this, it proves a curvature when carried out, the middle rod is usually around 3ft up, another argument i've not heard for a long time
20. plane curvature - i'm assuming you mean being able to see a curvature from a plane?
the perfect analogy is: put an ant on a boulder, he'll perceive it as flat, lift him up an inch, he'll still perceive it as flat
..but does that mean its flat? absolutely not, you're not high enough up, plain and simple, you're teeny weeny in comparison, the earth is frikkin ginormous!
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21. lack of video evidence - there's plenty, the US, Russia AND China have 24hr live feed of the earth spinning, they can claim its fake, but they can't claim it doesn't exist
22. photo manipulation - nasa openly admit that most pictures of earth are 'composites' due to the manner of how they're taken, there's original photos, again, not from just nasa, of the earth.. so FE's can scream and shout fake all they like, but i personally don't see the logic behind ASKING FOR PHOTO EVIDENCE FROM NASA, i've heard FE's before saying things like 'why don't they just turn hubble round and take a photo of earth?' well i'm sure they have their reasons but not a single flat earther would believe it anyway.. CGI!!! I do try to leave nasa out of my arguments though, I only mention them when people say things like you quoted; 'there's no photos' well there is, they just claim they're fake, there's a big difference
23. I've watched the full interview with Admiral Byrd, have you?
he said MANY times 'the continent of Antarctica', he reported masses of natural resources, he expected many more expeditions to recover the resources but instead.........
24. .....the Antarctic Treaty was signed and the massive ice shelf was renamed 'The Rockefeller Plateau' - to anyone with any real knowledge, that should explain a hell of a lot
25. fake moon landings? i couldn't give a shit to be fair, we've clearly sent something there at some point due to the advanced reflector test (not the old one that produced less accurate results)
26. i'm not really here for a discussion about the moon landings, again, that's a totally different topic
27. stars dont exist??? - come on now, lets not get too ridiculous, i look out of my window, i see stars, i need more than 'they don't exist', i've more credibility claiming headaches don't exist
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as for the religious angle, of course people thousands of years ago just assumed it was flat, in all fairness its completely understandable, we'd all assume it was flat without being told otherwise, again the ant metaphor
but we gained the knowledge to be capable of great things, the religious flat earthers should be worried about meeting the big guy at the pearly gates and having to explain how they were trying to belittle his work right..? I don't care who did it to be honest, god, aliens, or if we're just living in a cell within another living being.. its an absolutely wonderfully beautiful place, you guys are missing out, really you are
i'm here to ask questions and have a discussion.. i've done tests, i've cross examined..the earth IS a globe.. there's an abundance of evidence, from nasa, russia, china, amateurs, scientists, astronomers, to tests you can do yourself.. some things flat earthers can just claim as fake, some they literally can not, thats where cognitive dissonance plays a big part, well, maybe some are just very guillable
the flat earth on the other hand has absolutely zero actual physical evidence, zero
the OTHER theories, in my honest opinion, are simply created so the flat earther doesn't have to admit defeat, 'emporers new clothes syndrome', they cannot bring themselves to admit they were wrong, I don't mean that directly at you, i'm fully aware there's MANY theories.. some are just innocently taken in by it all, others are more 'I won't be beaten'
look, i believed it for a short while, ignorantly, without cross examining anything, but when i started looking into religions and their history, i realised you HAVE to cross examine everything, doing that provides better understanding of things..
cross examining evidence regarding the flat earth theory leads to one answer only.. the earth is a globe