Piston/Table

Block	Effect (Java Edition)	Effect (Bedrock Edition)
Barrier Beacon Bedrock Calibrated Sculk Sensor Command Block Creaking Heart Crying Obsidian Enchanting Table End Gateway End Portal End Portal Frame Ender Chest Grindstone Jigsaw Block Jukebox Light Block Lodestone Monster Spawner Moving Piston Nether Portal Obsidian Piston (extended) Piston Head Reinforced Deepslate Respawn Anchor Sculk Catalyst Sculk Sensor Sculk Shrieker Structure Block Trial Spawner Vault	Cannot be pushed.
Test Block Test Instance Block	Cannot be pushed.	N/A
Allow Deny Border	N/A	Cannot be pushed.
Compound Creator Element Element Constructor Hardened Glass Hardened Glass Pane Heat Block Lab Table Material Reducer Underwater TNT		Can be pushed.
Underwater Torch Colored Torches		Breaks when pushed, turning into drops when applicable.
Barrel Beehive Bee Nest Blast Furnace Brewing Stand Chest Chiseled Bookshelf Conduit Copper Chests Crafter Daylight Detector Dispenser Dropper Furnace Hopper Lectern Shelf Smoker Trapped Chest	Cannot be pushed, because these blocks hold block entities.	Can be pushed.
Large chest Large trapped chest Large copper chests		Can be pushed, but separates into two chests.
Banners Hanging Signs Signs		Breaks when pushed or unsupported, turning into drops when applicable.
Campfire Soul Campfire		Breaks when pushed, turning into drops when applicable.
Structure Void	Breaks when pushed, turning into drops when applicable.	Cannot be pushed.
^{[note 1]} Dead Coral Dead Coral Fans	Can be pushed, but breaks if unsupported, turning into drops when applicable.	Breaks when pushed or unsupported, turning into drops when applicable.
^{[note 2]} Carpets (not moss) Rail Powered Rail Activator Rail Detector Rail	Can be pushed, but breaks if unsupported, turning into drops when applicable.
Bell Copper Lantern Lantern Soul Lantern	Breaks when pushed, turning into drops when applicable.	Can be pushed, but breaks if unsupported, turning into drops when applicable.
Candles	Breaks when pushed, turning into drops when applicable.	Can be pushed.
Air Amethyst Cluster Bamboo Beds Budding Amethyst Button Cactus Cake Carved Pumpkin Cave Vines Chorus Flower Chorus Plant Cobweb Cocoa Copper Golem Statue Copper Torch Dead Bush Decorated Pot Doors Fire Fern Flower Pot Flowers Frogspawn Glow Lichen Heads Item Frame^{[note 3]} ^{[note 4]} Jack o'Lantern Ladder Lava Leaves Lever Lily Pad Melon Melon Stem Moss Block Moss Carpet Mushrooms Nether Wart Painting^{[note 5]} Pale Hanging Moss Pale Moss Block Pale Moss Carpet Pink Petals Pointed Dripstone Pressure Plates Pumpkin Pumpkin Stem Redstone Comparator Redstone Dust Redstone Repeater Redstone Torch Resin Clump Saplings Sculk Vein Sea Pickle Short Grass Shulker Box Snow Soul Fire Sugar Cane Torch Torchflower Torchflower Crop Tripwire Tripwire Hook Turtle Egg Twisting Vines Vines Water Weeping Vines Weighted Pressure Plates Wheat	Breaks when pushed, turning into drops when applicable.
Anvil	Can be pushed, but only when in a falling state.	Can be pushed, but falls if unsupported.
Concrete Powder Gravel Red Sand Sand	Can be pushed, but falls if unsupported.
Scaffolding Suspicious Gravel Suspicious Sand Dragon Egg	Breaks when pushed, unless if in a falling state.

Pistons cannot move blocks that require a support block, as they break and drop as an item (when applicable). There are however some exceptions, listed down below.

Exceptions

Carpets: since they consider all blocks excluding air as valid supports, moving carpets with pistons without them breaking is fairly easy.
- A carpet can even stay on top of a moving block without breaking.

Rails: they can be moved as long as they remain on top of a block's full face (if it has one) in their new position. Once the piston finishes to push or pull the rail, the block below can only be a moving block if its collision box is at the final position, otherwise the rail does not consider it a valid support.
- When the rail and the block supporting it are on two vertically parallel extending pistons, the rail doesn't break. Trying to move the rail and its support block with perpendicular pistons causes the rail to break (although in Java Edition, due to the bug MC-75716, it visually appears to not break). In Java Edition, the piston moving the block can be powered 1 block event after the piston moving the rail in order for the rail to be moved reliably without breaking.
- Rails re-orient themselves after being pushed, similar to when placed manually.

Dead coral and dead coral fans: in Java Edition, they can be pushed similarly to how rails can; however in some situations the coral will both break and get moved by the piston, emitting block updates at its original position.
- This mechanic is frequently used in TNT dupers.

References

↑ Check the exceptions section
↑ Check the exceptions section
↑ In Java Edition, item frames are entities, not blocks. In Bedrock Edition, they are blocks.
↑ In Java Edition, if the "Fixed" NBT tag is set to "1", the item frame does not break when attempting to push it using a piston, but it still does not push. If the "Invulnerable" NBT tag is set to "1", the item frame breaks when pushed.
↑ Paintings are technically entities, not blocks.

[1] Check the exceptions section

[2] Check the exceptions section

[3] In Java Edition, item frames are entities, not blocks. In Bedrock Edition, they are blocks.

[4] In Java Edition, if the "Fixed" NBT tag is set to "1", the item frame does not break when attempting to push it using a piston, but it still does not push. If the "Invulnerable" NBT tag is set to "1", the item frame breaks when pushed.

[5] Paintings are technically entities, not blocks.

[note 1]

[note 2]

[note 3]

[note 4]

[note 5]