Mod edit:
When the player stands at the very edge of a block so that the center of the player is over the edge of the block, the game calculates the actions of the player according to the (air) block below the center of the player but not to the block the player is standing on.
walking completely silent
walking speed on Ice edge is normal
fall damage when falling on the edge of slime blocks
no reduced fall damage when falling on edge of hay bale
wrong/no fall particles when landing on the edge of a block (allows for MC-240028, MC-240029 and MC-122547 to be seen)
no damage while standing on the edge of magma blocks
can jump normally on the edge of honey blocks
jumping while standing on the edge of farmland does not turn it into dirt
redstone ore not activated when standing on the edge
not being affected by soul speed when standing on the edge of soul sand/soil
when landing on wool with the player center above air, sculk sensors are activated (MC-252389)
when walking on the edge of blocks, sculk sensors do not detect vibrations (MC-207290)
taking fall damage at the edge of powder snow
Walking at the very edge of a block, and I mean the longest out possible on a block, will make it silent since it is not registering the walking. You are still on the block, just that it is silent. This bug has existed since before alpha.
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Confirmed, this occurs on mac computers too.

Please don't mark an issue as affecting a future version; you can't have tested it already 🙂

Confirmed in 1.4.6.

I think the problem is that when you walk out oh the very edge of a block, you're actually no longer on the original block. If you go into 3rd person mode, you can see that you aren't actually on the block, and that you're actually on the air. It still registers that you are standing on the block, but not the part that makes noise.

Still a concern in 1.7.5, 14w08a and 10b, relates to MC-44381

still in 10c

Confirmed in 14w28b. It does have to do with the fact that you are not technically "standing" on the block. I will upload a couple pictures of the F3 menu showing this

These are two screenshots showing the bug. Please note the x values of "Block" and "Looking at". The ice is along x=1, but the "Block" value you are "standing" on is 0.

Confirmed in 1.8.1-pre2.

Confirmed for 1.8.8 & 15w33c.

Still happening in 15w42a.

Confirmed for 15w43c.

Confirmed for 15w44a.

Confirmed for 15w45a.

Confirmed for 15w46a

Minecraft 1.8.9 and 15w51b confirmed

Confirmed for 16w02a.

Confirmed for 1.9.1-pre3.

Confirmed for 1.9.3-pre3.

Confirmed for 1.9.4.

Confirmed for 16w20a.

Confirmed for 16w21a.

Confirmed for 16w21b.

Confirmed for 1.10-pre1.

Confirmed for 1.10-pre2.

Confirmed for 1.10.

Confirmed for 1.10.1.

Confirmed for 1.10.2.

Confirmed for 17w17b

Confirmed for 1.12.2.

Affects 18w14b

Affects 18w15a

Affects 18w16a

Affects 18w19a

Affects 18w19b

Affects 18w20a

Affects 18w20b

Affects 18w20c

Affects 18w21a

Affects 18w22a

Affects 18w22b

@unknown, ticket is yours now

Affects 18w22c. Can a relates to be added for MCPE-18063?

It seems as though sprinting on the edges of water no longer shows water particles.

Water particles probably don't show up with this bug because water is no longer really a block and is instead more of just a block tag.

Then why does it still show up upon falling?

This issue makes falling on slime blocks adjacent to air quite risky

This makes descending very dangerous! (see MC-139088)
When you're standing on the edge of a scaffolding tower and try to descend, you'll actually fall and take damage.
Video link (google drive): https://drive.google.com/file/d/1Pmc4Y5UvmTm5ncG6isO6HNf6qH_lztGu/view?usp=sharing

This bug allows you to stand on the edge of a magma block and not take any damage.
Also affects scaffolding.

Also affects honey blocks.

Should this ticket be split into multiple smaller tickets? Granted they all originate to this same cause though, so maybe not.

I don't think it would be useful to split this one, since there is one underlying cause that, if fixed, would be fixed for more or less all of the cases. (It would require pretty systemic changes, but also almost certainly wouldn't be fixed for only one of the cases without fixing the rest.) Compare to, say, MC-9553, where the bug is composed of lots of individual interactions that each needed to be fixed individually; there, splitting made sense.

Allows you to jump onto iron block in attached picture if you stand on edge of honey block
[media]
This issue also affects blocks that have the climbable tag like for example ladders, vines. I am not sure if this is intended for these blocks but I believe that it's not.
Soul Speed is affected as well, see MC-174483 for instance.

I did some testing planning to open my own issue specific to honey blocks, but luckily found this one so I do not duplicate. But I will put what I found here anyway.
Java Minecraft 1.15.2
I put a honey block at -56, 56, 29. When crouch walking off the edge a player (and I assume any mob) can jump as though they were not on a honey block. I teleported myself right to the very edge of the block, at -55.88, 57.00, 29.999. (landing at y = 56.93750 if that matters at all) At that location the player's jump IS affected by the honey. But then I teleported to -55.88, 57.00, 30.00. At this location the player can jump freely uninterrupted from the honey block.
I did some more testing. I discovered some unrelated facts, like that the coordinates 29.99999999999999, 29.999999999999993, and 29.999999999999996 are all valid, but all other coordinates in between and after are rounded. Also all of these coordinates on the honey block from the last paragraph still affect the player's jump height as though you are an the honey block, so all is working as intended here.
I would suggest the player's hit box determine if it is affected by a block or not. This could break things in ways that I couldn't possibly think up, but obviously not all of the physics in this game work based on what is directly below Steve. If it did, we couldn't crouch walk off the edge of a block and hang onto it. I imagine that the same bit of code that tells the game not to fling Steve to the ground if you manage to get your center of mass off a block by .01 could help us out here. But I am only learning Java. Not yet worthy to be called a programmer, so I don't really know.

Confirmed for 20w18a. This also affects magma blocks, but not damage of campfires. (normal and soul)

Awesome! This feels so sneaky!

Confirmed for 20w20b

Striders also get cold standing on the edge of falling lava. They also don't seem to travel upwards anymore. Think this also relates to this issue.
[media]

That should be MC-181610. If you find a case where the center of the strider actually matters, please create a new report and say here that you did, so that it can be marked as related.

This bug also affects hay bale, so when you land on the edge of hay bale, it won’t reduce fall damage.

Confirmed for 20w21a

Confirmed for 1.16 Prerelease 1

Confirmed for 1.16 Pre-2

Confirmed for 1.16 pre-release 7

Also, only farmland that is under the center of the player gets turned into dirt when jumping.

Confirmed in Snapshot 20w45a.
Can confirm in 20w48a.

This also affects cobwebs and my guess it also affects powdered snow although I haven't tested it
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Also, this must be the oldest bug report still not resloved. Correct me if I am wrong
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Also, confirmed for 1.16.4

There's no reason it wouldn't affect literally every block, if we understand it correctly. The issue is in the code that calculates whether a player is "on" a block or not for the purposes of any effect relating to standing on a block.
This is a major bug but it's also likely an extremely nontrivial fix; not only would the devs need to change the code that calculates whether a player is standing on a given block, they'd also need to add code to handle any potentially conflicting effects (such as if a player is standing on both soul soil and honey blocks while wearing soul speed boots, for example). While this could absolutely be done, there's no super obvious answer for how to go about it. Personally, what we'd do is just have the effects stack, and cancel out if they conflict? But that also means they'd have to make sure standing on the line between two blocks with the same effect doesn't cause the effect to be doubled!!
I do hope mojang is at least discussing how to handle this, though. It seems like something very much worth prioritizing once other major bugs have been handled.

In 20w49a, when Sculk Sensors were introduced, it was revealed that you are able to land on air, which triggers a Sculk Sensor. Best demonstrated when landing on a wool block, but your hitbox is more than 50% off the block.

I suggest either trying to make the hitbox the player's shape so its a bit more accurate or to limit the distance that you can sneak.
Can confirm in 21w05a.
Can confirm in 21w05b.
This can now happen with powder snow when you're wearing leather boots.

Powder snow is like climbing powder snow when wearing leather boots works like scaffolding, that is why powder snow is affected by MC-139063.
This bug does not affect big dripleaf.

Confirmed in 1.17 pre-release 4
This bug also amusingly affects light blocks (landing on the corner of a solid with a light block under your center) causing yellow fall particles

Another effect is walking on the edge of powder snow doesn't slow you down, showcased in MC-227191

Can confirm in 1.17.1 Pre-release 3.

Can confirm for 1.17.1.

This also prevents soul sand from slowing you down.

Affects 1.18.1

Can confirm as of 22w03a

Affects 1.18.2

How many People Tested this! it has like 200 Affects Versions lol and can confirm for 1.18.2.

Yeah, this has been a thing since beta (probably further back, but I haven't played earlier versions)

Code analysis (tentative)
This is probably because checks like this look directly below the center of the player to find what block the player is "standing" on. Instead, collision should track what blocks the player is colliding with and use that block instead.

In 22w17a.

MC-254644 might be related
Can confirm in 1.19.2.

Can Confirm for 1.19.3 Release Candidate 3

Can confirm in 23w03a

Can confirm in 23w04a

Can confirm in 23w05a

Can confirm in 23w06a

Still not completely resolved in Minecraft 1.20 Pre-release 6 with scaffoldings. Scaffoldings are not included in the bug report. A duplicated bug report is MC-165067
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With at least 4 scaffoldings, go to the top and hold shift at the edge of it to get fall damage.
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Video of the bug being played in Minecraft 1.20 Pre-release 6: bug video

@unknown That is MC-139063, which is a separate bug.

Relates to MC-269734

Relates to MC-274947