Statue

Statues can be found all throughout the Dungeon. Most of them are harmless, but some contain dangerous traps or actively try to kill you. While each kind of statue behaves in a unique way, they are most easily divided into inanimate statues (which are treated like dungeon features) and animated statue monsters (which aggressively attack you and can be killed in turn). Whether animate or not, all statues are completely immobile, and if a statue is capable of attacking you, the game will auto-exclude all tiles in their LOS to keep you you from accidentally wandering into danger.

All statues behave much like transparent walls, being a barrier to fire but not visibility; unlike transparent walls, they are not a barrier to reaching or smite-targeted attacks. They are not affected by Dig or a wand of digging. Statues do not regenerate HP.

Monster Statues

Most animate statues have elemental resistance or immunity. Some notable statues are susceptible to a wand of disintegration, while the rest are unaffected. All can be taken down by physical attacks or spells: eg, Lee's Rapid Deconstruction, Shatter, Orb of Destruction.

Archer statue - Shoots arrows, has limited ammo
Charred statue - Throws flames
Conjurer statue - Casts Orb of Destruction
Diamond obelisk - Surrounded by a constant Tornado effect, nearly indestructible
Firespitter statue - Uses Fire Magic to defend treasure vaults in the Volcano
Frost covered statue - Throws frost
Ice statue - Summons ice beasts, shoots frost and freezing clouds, vulnerable to fire
Lasher statue - Attacks with whip when you enter melee range
Lightning spire - Shoots electricity
Orange crystal statue - Inflicts Divinations miscast effects, vulnerable to disintegration
Pillar of salt - The unfortunate remains of demons or undead which suffered the wrath of Zin.
Roxanne - A statue unique with powerful Earth Magic
Silver statue - Summons demons, vulnerable to disintegration
Snail statue - An idol of Cheibriados that repeatedly casts Leda's Liquefaction
Spooky statue - Casts Haunt
Training dummy - A harmless, fragile statue found only in the Tutorial and a few vaults
Warrior statue - Hits in melee with a weapon, found in a vault which forces you into melee range
Wizard statue - Casts various spells

Inanimate Statues

Inanimate statues, while unaffected by physical attack, are susceptible to disintegration and spells with a physical component.

Granite statues come in a variety of shapes, often resembling dangerous monsters, but pose no risk; at worst they may prevent you from reaching a shop or pile of treasure.
Orcish idols are like granite statues, but destroying them will annoy Beogh. When annoyed, Beogh will smite you 100% of the time if you're one of his followers, 50% of the time if you're a hill orc, or 33% otherwise. Always look at the description of a statue before destroying it, just in case.
Scorch marked granite statues are just like granite statues, except that they are found in areas where flame clouds sporadically appear. They do not actually cause these clouds, so while you can destroy them, doing so accomplishes nothing toward making the area safer.
Elephant-shaped statues are inanimate lumps of stone occasionally found in the temple of Lair:8. They do nothing until you venture into the treasure vault in the back of the temple, at which point several of them will come to life and attack you as dire elephants. Be sure to close doors behind you as you explore; this will allow you to only face a few of them at a time.
Gravestonesare generally found in special vaults or portal levels (like Zonguldrok's Mausoleum, for example). If they show up in the main dungeon, it's a sure sign that the undead are wandering the level.

A nameless grave. The question remains: is it the grave that is nameless, or the person who lies in that grave?

Sarcophagi are found in The Tomb and behave like granite statues.

Immobile Monsters

There are also some monsters which are immobile and may create movement exclusion zones when discovered, but unlike statues they are not nonliving and regenerate HP when injured.

P Ballistomycete- Launches fast-moving spores to reproduce, and blow you to bits