Rimmington
Rimmington is the default when a house is bought. It only requires 1 Construction. It is easy to bank by walking east to the Squire in Port Sarim then going to the Void Outpost to bank.
Taverly
Taverly requires 10 Construction and 5,000 coins to move. It is not near a bank, but it is good for Amulet of Glory recharging, as it is near the Hero's Guild, and good for Summoning, as it is near the Obelisk below Pikkupstix's house.
Pollnivneach
Pollnivneach requires 20 Construction and 7,500 coins to move. It is no where near a bank, but it is very good for inner desert travel.
Rellekka
Rellekka requires 30 Construction and 10,000 coins to move. It is not very close to a bank, but it is useful if you forget your lyre and for switching to lunar spells, etc.
Brimhaven
Brimhaven requires 40 Construction and 15,000 coins to move. It is near the Brimhaven Dungeon and some other resources.
Yannille
Yanille requires 50 Construction and 20,000 coins to move. It is very close to a bank and is AMAZING for Prayer training.