The bug
Signs and carpet does not drop when the underlying block is moved by a piston.
This may be intended given that MC-17851 was fixed (though possibly not on purpose), however it is inconsistent with other blocks which require a supporting block. These blocks do drop when the supporting block is moved, even if the new block at that position would support them as well.
See the attached
[media]showing a setup for reproducing this and outlining the inconsistency.
Related issues
is duplicated by
relates to
Attachments
Comments

Duplicate of MC-3287.
Confirmed for 1.12.1
Confirmed for 1.13.1.

Cannot reproduce in 18w47b
@unknown, I still can reproduce for 18w47b.

I tested all possibilities more extensively, and I've come to a different outcome than I expected. While assuming the sticky piston is always setup to immediately push against a solid block, and the sign and carpet is attached to that block, the following happens.
When the sticky piston is placed to extend upwards before powering:
On extend, sign will stay but carpet moves up with the solid block
On retract, sign will still stay but carpet drops because it's not attached to a solid block anymore
When the sticky piston is placed and powered to extend upwards:
On retract, sign and carpet both will pop off
When the sticky piston is placed to extend sideways before powering:
On extend, sign and carpet both will stay put
On retract, sign and carpet both will stay put
When the sticky piston is placed and powered to extend sideways:
On retract, sign and carpet both will pop off
With these results, I'm not sure anything is broken here.
Still an issue in pre-7.

Removed the old screenshot attachments since they were in part describing MC-17851, which was slightly misleading. Instead attached a video showing the inconsistency.

Now following the video clip, I see what is wrong, and it still happens.
Can confirm in 1.18.1.
Can confirm in 1.18.2.
Can confirm in 1.19.
Can confirm in 1.19.2.

the reason for this is that carpets and signs do not require supporting faces, which the moving_piston block does not have. most other support blocks need full faces, so when the sandstone is turned into moving_piston, it no longer has a solid face to anchor to.