Signs and chests do not display cracks on their surface when being broken, like “normal” blocks (and fences and trapdoors!) do.
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But didn't chests display the cracks in the past?
Torabi, yep — chests used to be rendered as normal blocks. This changed in β1.8 (Adventure Update) which gave them an opening-lid animation and made them slightly smaller than full blocks.
yeah this is an old bug and i would like to see it fixed
I don't care if they've changed and that's why it doesn't have the animation, I want the animation back to make it look better.
This isn't an bug. Like Kevin said, the chests were full blocks before 1.8. Since then, they are considered transparent blocks now and can't display breaking animations.
Java just won't allow it.

That's ridiculous. Of course Java will allow it, just like it allows the lid-opening animation. The programming language itself is rarely a limitation. The question is just how much work it will take to make it work with the rest of the code.
The reason is signs and chests are not actually blocks but tile entities. And entities don't naturally display cracks.
Sure. I could retitle this bug “Tile entities should be capable of displaying the breaking animation”, but I think it's better to describe the end result, not the implementation.
Confirmed.
It is definitely a size problem, not a tile entity issue. Mob spawners and furnaces work just fine.
Still in 13w24a

Still in 1.7.2 and also with CocaBeans

Signs did have a breaking animation recently, though it was entirely due to a bug. See MC-32082. I think it is a problem of the way the block renders, from a sheet or in pieces.

Still in 14w05a

Partially fixed in 14w07a.
Signs and Chests have no animation
Cocoa Beans and Beacons have the animation

Still in 14w08a

Still in 14w10a/b
Also in 14w10c
that's because they don't get the textures from the blocks dictory like all other blocks do, signs and chest textures are under the entity's dictory and have the complete block in 1 png instead of 1 side of the block in 1 png like all other blocks
Confirmed for 14w11b
Confirmed for 14w17a
Confirmed for 14w18a

Confirmed for 14w28a.
Is there any way we can get a Mojang employee to confirm whether this is a bug or a Works as Intended?
Also, confirmed for signs, chests, and trapped chests in 14w28b
Confirmed for 14w28b

Banners are not showing breaking animation: MC-63509
That is because of the way the texture is rendered.