When I entered this village, I saw water comming out from the farm like there was a hole in the farm but there isn't.
See for yourself
seed: 5481915211951425789
x = 89
y = 52
z = 28
Flatland setting: 2;7,50x3,2;1;village
@unknown's possible explanation and what is important to know to reproduce this
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Please provide seed and coordinates.

seed: 5481915211951425789
x = 89
y = 52
z = 28
Flatland setting: 2;7,50x3,2;1;village
Confirmed. (/me waits on Markku's comment on this one)
Also confirmed. Water disappears on block update.

Took quite a while to get my Eclipse accept the latest MCP (a setting was messing with JRE detection, had to figure that out first), and apparently the MCP result for 1.5 was sort of without proper field and method name deobfuscation. I guess it is a bit much to expect them to be done so soon after release 😛
Anyway, this seems to be an elusive one. My first idea went down the drain, and the real cause is a mystery. However, I think I managed to get a workaround. The farm generation likes to use flowing water (no idea why or how they turn into source-blocks). I switched those to place directly source blocks, and could not reproduce this issue.
The already "leaking" things do remain in place even after the fix. This was expected and just means the state has been saved to the world and is not just some runtime bug or weird rendering issue.
I will delay deeper study on this until I get a bit more mature version of MCP. Hopefully Mojang will tackle it before that, as they can use change history etc. to speed up tracking potential causes.
happened to me to. I used Superflat though and it was at two villages. Seed:-60146107116314948 X:-1158 Y:14 Z:332 and X:-900 Y:9 Z:338
Added a screenshot of the village I found on a default superflat world I created.
Not just superflat worlds, here's one from largeBiomes. MC Version snapshot 13w24b. Seed 2028444982885494103, x=803, y=74, z=84.
I see this all the time.... also when taking away water sources, sometimes a corner acts as a source block

I started getting this on Superflat worlds after the last Snapshot.
I see tons of duplicate reports, but none of them return to vote. I suspect this issue won;t be touched for months.

I just opened a new world in 1.6.1 and saw this issue.

Is this still a concern in the current Minecraft version? If so, please update the affected versions in order to best aid Mojang ensuring bugs are still valid in the latest releases/pre-releases.

Still in 1.6.2
happens with lava in blacksmith house's too. see images in MC-26697

Probable reason
Going through those few cases where players have bothered to provide seeds, settings and locations, this seems to be a side-effect from the way "decorations" are added to world - shifted by half a chunk. Thus, when farms (or any other structures) are generated and inserted into the world, they can be inserted partially (cut at half-chunk positions), the other part(s) of the structure being inserted later, when player moves and allows generating more chunks. If the cutting happens to be next to a flowing block, there is half a chunk more space to flow into. Later when player moves, the other parts will overwrite some of the flowing water blocks without mercy, leaving the remnants as artifacts reported here.
Note, normal water generation is done per full chunk, and uses "still water" block, instead of "moving water" block. I do not know why, but in changing the village farm water blocks into "still water" type seems to fix the symptoms. Or more like, it is a workaround, since the real reason is the half a block shift (which causes more issues than just this one). Likely, the same could be applied to the "moving lava" used in one house, however, I do not have suitable test case for that.
A proper fix might be to implement some additional code that delays/prevents planting a structure (or a full independent part of a structure, depending on terminology) into the world until all of it can be planted at once, but that is beyond the scope of a comment here. However, this approach can have other nasty issues due to the way how some decorations will then be possibly planted when they shouldn't be, as they won't know about the delayed structure. (That last part is the roughly reason e.g. why lakes sometimes carve tree or building bottoms out; the trees/buildings do not know about the lake coming later...)
Here is example of workarounds, for those who want to test if it really works for them, too:
ComponentVillageField.addComponentParts()
//this.fillWithBlocks(par1World, par3StructureBoundingBox, 3, 0, 1, 3, 0, 7, Block.waterMoving.blockID, Block.waterMoving.blockID, false);
//this.fillWithBlocks(par1World, par3StructureBoundingBox, 9, 0, 1, 9, 0, 7, Block.waterMoving.blockID, Block.waterMoving.blockID, false);
this.fillWithBlocks(par1World, par3StructureBoundingBox, 3, 0, 1, 3, 0, 7, Block.waterStill.blockID, Block.waterStill.blockID, false);
this.fillWithBlocks(par1World, par3StructureBoundingBox, 9, 0, 1, 9, 0, 7, Block.waterStill.blockID, Block.waterStill.blockID, false);
ComponentVillageField2.addComponentParts()
//this.fillWithBlocks(par1World, par3StructureBoundingBox, 3, 0, 1, 3, 0, 7, Block.waterMoving.blockID, Block.waterMoving.blockID, false);
this.fillWithBlocks(par1World, par3StructureBoundingBox, 3, 0, 1, 3, 0, 7, Block.waterStill.blockID, Block.waterStill.blockID, false);
I had this a few days ago in my 15w49b server, going to provide seed and coords later, as I'm not on my pc now
Edit:
ok, weird, I generated a world with the same seed, got to the same coords, and now it wasn't there

IIRC, this bug was back then a slightly randomly appearing, as it depended in some cases on direction in which the user approached the village from, the exact chunks traveled (thus exact order of chunks generated), or even some timing related issues. Especially, warping directly on the village may fail to show the issue. It is best to teleport just outside the generation range of the village (note, it might not be the same as rendering distance), note which direction the approach is, see if issue happens. Note, even small (one chunk) differences in the approach (and thus chunk generation order) could affect the result.
To search for cases, it is best to just fly around in superflat, instead of regenerating the same world or same village. Only once the issue shows up, record the direction of movement and village location, and try to repeat with the same direction & village.
Happy leak hunting.

Please include @unknown's possible explanation and what is important to know to reproduce this

Does this still occur in the current versions?
Is this still an issue in the latest snapshot 16w44a?

Is this still an issue in the most recent versions (currently that is 1.11) of Minecraft? If so, please update the affected versions and help us keeping this ticket updated from time to time. If you are the owner/reporter of this ticket, you can modify the affected version(s) yourself.

Since 1.8 at least, worldgen has been placing Stationary water and not Flowing, so that part of Markku's analysis (as he called it, the 'workaround') ended up not being effective as a solution.