There are many homes the player can aquire in LoreRim. Some are acquired from quest, some are bought and some are built. See more details below
In the following cities, all the way back in 2011 we had 5 player homes we could buy. (Plus one from Dragonborn DLC)
Whiterun - This can be purchased from Proventus Avenicci after “Bleak Falls Barrow” quest.
Riften - Purchased from Anuriel after becoming Thane of the Rift.
Purchased from Raerek or Reburrus Quintilius after becoming Thane of the Reach.
Solitude - Purchased from Falk Firebeard after becoming Thane of Haafingar.
Windhelm - Can be purchase from Jorleif after completing the “Blood on the Ice” quest and becoming Thane of Eastmarch.
Raven Rock - Complete the quest “Served Cold” to be awarded the home
The hearthfire dlc added 3 player homes which the player would build from scratch and had some customisation options which allowed the player to make the perfect player home.
Falkreath Hold - Purchase the land from Falkreath steward after helping the Jarl (you will recieve a letter from a courier)
Hjaalmarch - Purchase the land from Morthal steward after becoming Thane of Hjaalmarch.
The Pale - Purchase the land from Dawnstar steward after becoming Thane of the Pale.
The following player homes were added by Creation Club and are included in the anniversary upgrade (which is a requirement for Lorerim).
Riften Ratway -Purchase from Vekel the Man in the Ragged Flagon.
East of Whiterun - Purchase the deed from the steward in Dragonsreach.
The Reach - Complete the quest “Hendraheim” which starts when you recieve letter after completing the "The Silverhand" quest.
Hjaalmarch - Unlocked by completing its modded quest which starts on discovery
Blackbone Isle Grotto - Complete the “Dead Man’s Dread” quest to unlock the ship as a home.
The Rift - Complete “The Sanctuary and the Manufactory” quest started by reading Seeks-Ancient-Artifacts’ Journal in The Frozen Hearth
Northern shore of Mara’s Eye Pond - Gallows Hall will remain locked until you've completed the Conjuration Ritual Spell quest.
Near Rorikstead - Complete “The Unquiet Dead” quest started by asking innkeepers "Are there any problems around town that need handling?"
Bloodchill Cavern - Complete the “Guests for Dinner” quest, you'll receive your invitation after completing the "Bloodline" quest.
Each of the cities in Lorerim have been overhauled and included in those changes are several new player homes. A big change is the smaller capitals: Falkreath, Dawnstar, Morthal & Winterhold all have homes in the actual cities themselves, not just some land in the hold you can but. Additionally, the some of the other hold capitals have new homes added alongside the vanilla ones.
Winterhold - Purchase from Winterhold steward (requires Thanehood)
Morthal - The key is inside a satchel in Movarth’s Lair
Dawnstar - The key is inside a satchel in Nightcaller Temple, in the dorms
Whiterun- Purchase the deed from Quintus at the market place
Windhelm - Purchase the deed from Sadri at Sedri’s Used Wares for 1,000 septims
Riften - Unlocked by becoming Guildmaster of the Thieves Guild
Skaal Village - Key can be found in a knapsack at Snowclad Ruins by the altar with dead animals
Kynesgrove - Bought from a nearby NPC
Solitude - Bought from innkeeper of the Winking Skeever
Some new towns were added in Lorerim which came with some player homes. Additionally the smaller towns and villages that already existed have been overhauled which now have a player home.
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Amber Guard - Purchase via sign next to home
Granite Hill - Obtained after solving the mystery plaguing the town (requires starting main quest)
The Rift east of Riften - Bought from Maven Black-Briar after you become Thane of Riften and completed the quest Promises to Keep.
Last Vigil - Defeat the Ebony Warrior
Wyrmstooth - Purchase an abandoned imperial fort from Lurius Liore after completing Wyrmstooth's main quest
Whiterun Hold - Just walk in and kill the N'wah, then the house is yours