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)
Breezehome (Whiterun) - This can be purchased from Proventus Avenicci after “Bleak Falls Barrow” quest.
Honeyside (Riften) - Purchased from Anuriel after becoming Thane of the Rift.
Vlindrel Hall (Markarth) - Purchased from Raerek or Reburrus Quintilius after becoming Thane of the Reach.
Proudspire Manor (Solitude) - Purchased from Falk Firebeard after becoming Thane of Haafingar.
Hjerim (Windhelm) - Can be purchase from Jorleif after completing the “Blood on the Ice” quest and becoming Thane of Eastmarch.
Severin Manor (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.
Lakeview Manor (Falkreath Hold) - Purchase the land from Falkreath steward after becoming Thane of Falkreath.
Windstad Manor (Hjaalmarch) - Purchase the land from Morthal steward after becoming Thane of Hjaalmarch.
Heljarchen Hall (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).
Shadowfoot Sanctum (Riften Ratway) - Purchase from Vekel the Man in the Ragged Flagon.
Tundra Homestead (East of Whiterun) - Purchase the deed from the steward in Dragonsreach.
Hendraheim (The Reach near the border of Whiterun & Falkreath) - Defeat the warrior during the quest “Hendraheim” and claim the home which starts when you recieve letter from courier after completing the companions quest "The Silverhand".
Myrwatch (Hjaalmarch near Morthal) - Claim by discovering the tower and completing the puzzle to unlock the door.
Dead Man’s Dread (Blackbone Isle Grotto) - Complete the “Dead Man’s Dread” quest to unlock the ship as a home.
Nchuanthumz (The Rift) - Complete “The Sanctuary and the Manufactory” quest which is started by reading Seeks-Ancient-Artifacts’ Journal, v1 in Frozen Hearth inn within Winterhold.
Gallows Hall (Northern shore of Mara’s Eye Pond) - Gallows Hall will remain locked until you've completed the Conjuration Ritual Spell quest.
Goldenhills Plantation (Near Rorikstead) - Complete “The Unquiet Dead” quest upon entering the farm.
Bloodchill Manor (Bloodchill Cavern southwest of Winterhold) - Complete the “Guests for Dinner” quest, you'll need to complete the "Bloodline" quest in the Dawnguard DLC before being receiving your invitation to dinner
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.
Frostview Hall (Winterhold) - Purchase from Winterhold steward
New Moon Cottage (Morthal) - The key is inside a satchel in Movarth’s Lair
Breaking Dawn Cottage (Dawnstar) - The key is inside a satchel in Nightcaller Temple, in the dorms
Bluesky Hall (Whiterun) - Purchase the deed from Quintus at the market place
Graystone Lodge (Windhelm) - Purchase the deed from Sadri at Sedri’s Used Wares for 1,000 septims
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.
Vintrhus (Skaal Village) - Find the key in a knapsack at Snowclad Ruins by the altar with dead animals
Kynesby (Kynesgrove) - Bought from a nearby NPC
Cliffside Manor (Amber Guard) - Purchase via sign next to home
Crossway Cottage (Granite Hill) - Obtained after solving the mystery plaguing the town
Black Briar Lodge (The Rift east of Riften) - Bought from Maven Black-Briar after you become Thane of Riften and completed the quest Promises to Keep.
The Ebony Keep (Last Vigil) - Defeat the Ebony Warrior
Fort Valus (Wyrmstooth) - Purchase an abandoned imperial fort from Lurius Liore
Clockwork Castle (Velothi Mountains) - Complete the clockwork quest mod started by entering the mountain pass on the easten side on the map in eastmarch
Project AHO (Sadrith Kegran) - Complete the Project AHO’s main quest (depends on the ending you get)