here are 27 Pyromancies in the game. Most can be bought from vendors (some of which require finding a Pyromancy Tome), some are found in a specific location, some are traded for boss souls, and one (Warmth) is a covenant reward. Vendor and Pyromancy Tome information can be found at the end of the solution.

Fireball
Sold by Cornyx.

Fire Orb
Sold by Cornyx after returning the Great Swamp Pyromancy Tome.

Bursting Fireball
Sold by Cornyx after returning the Great Swamp Pyromancy Tome.

Great Chaos Fireorb
Sold by Cornyx after returning the Izalith Pyromancy Tome.

Chaos Bed Vestiges
Trade with Ludleth of Courland for the Soul of the Old Demon King.

Fire Surge
Sold by Cornyx.

Fire Whip
Sold by Karla after returning the Quelana Pyromancy Tome.

Firestorm
Sold by Karla after returning the Quelana Pyromancy Tome.

Chaos Storm
Sold by Cornyx after returning the Izalith Pyromancy Tome.

Great Combustion
Sold by Cornyx.

Sacred Flame
On a corpse in a lava pool just beyond where you fight Knight Slayer Tsorig in the Demon Ruins area of the Catacombs of Carthus.

Profaned Flame
From the Irithyll Dungeon bonfire, take the shortcut door to the right and ride the lift down to the bottom. The spell is on a corpse next to the Giant.

Poison Mist
Sold by Cornyx after returning the Great Swamp Pyromancy Tome.

Toxic Mist
Found in a lava pool in Demon Ruins area of the Catacombs of Carthus.

Acid Surge
Sold by Cornyx after returning the Carthus Pyromancy Tome.

Flash Sweat
Sold by Cornyx.

Profuse Sweat
Sold by Cornyx after returning the Great Swamp Pyromancy Tome.

Iron Flesh
From the Farron Keep bonfire, go outside and head right. You'll see it in the swamp among some trees.

Power Within
Found on a corpse behind a hidden wall on the first level of the Grand Archives that is accessed by pulling a lever in a somewhat dark room just beyond the first wax pool.

Carthus Beacon
Sold by Cornyx after returning the Carthus Pyromancy Tome.

Carthus Flame Arc
Sold by Cornyx after returning the Carthus Pyromancy Tome.

Warmth
Reward for reaching rank 2 in the Mound Makers covenant. This requires 30 Vertebra Shackles, which are rewarded for successful Mound Maker invasion kills and are very rare drops by Red-Eye Skeletons in the Catacombs. This was my farming technique: