We used to call it Worst Asheville. Back in the day West Asheville was pretty run down. Boy, have the tides shifted on this neighborhood! Today, West Asheville is one of the most expensive areas in Asheville to buy a home. For good reason though. It's a great area to live if you like to walk. Characterized by the 1920's bungalows and little cottages, west Asheville for the most part is very pedestrian friendly, very family friendly, and eclectic.
What I like about west Asheville: Tree-lined streets, eclectic and hip neighbors, walking to great bars, pubs and restaurants. Bike friendly to downtown through the River Arts District. Easy access to the French Broad River.
There are a number of distinct areas in West Asheville, Malvern Hills, East/West, etc.
Leicester is also included in the 28806 zip code, but this area has a more country vibe.