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- pressure plates require the block below to have a rim (e.g. hopper) or center point on top (e.g. glass pane)
- rails require the block below to have a rim
- banners and cakes require the supporting block to be a solid material (materials should probably be documented somewhere)
- bells and levers require the supporting face to be "full"
- carpet just needs to be above anything but air
- lanterns need a center point on top (if place on top) or below (if placed below) -- for example a lantern can hang below a down-pointing hopper but can't be placed on top
- redstone dust requires a full face below, but is also hardcoded to allow hoppers

The "rim" and "center point on top" are apparently well-defined in-game predicates; for example, the center point is a 2x2 center square on the hitbox (not visual box) assuming the surface is a 16x16 square grid. Quoting tryashtar on the "rim", "It's a predicate that is created specifically by removing the entire center section from a full cube, leaving only the, well, rim".

In references to other placement quirks (by tryashtar):

Sea pickles just check if the below block has a non-empty upward face
Not a block but item frames are really interesting, they can be placed anywhere that doesn't intersect with their hitbox (and support block material must be solid (or a repeater/comparator))

By the same person, on leaves' exceptional behavior:

Leaves are considered to have faces that are "full" but not "sturdy"

Rigid blocks (blocks that require a rim):

  1. Pressure plates can also be placed on blocks with only a center
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