--Montford Neighborhood-- I lived in this neighborhood 3 years ago & loved it! I still sometimes go over there just to go for a walk. There are two community gardens, a park, a community center with a climbing wall, huge neat old houses split into apartments & newer faux historical developments. Single resident houses & Bed & breakfasts are also everywhere. Someone just opened a fancy little Jamaican restaurant in the heart of Montford as well.
--South French Broad road area & west end neighborhood are hit or miss. I would live in this area but I only suggest it for braver folks. Theres a crappy old public pool that I like cause noones ever around & the YWCA there is super nice. Theres a great student acupuncture clinic centrally located in a big old house. Everyone I know that lives in this area have either a burglar alarm or not much a person would want to steal. Its walking distance to downtown.
--I wouldn't bother checking out Shiloh in south asheville & don't live near pisgah view apartments in west asheville!
--West asheville near haywood rd. is actually pretty great. Super neighborhood feel & full of bungalows. Central to this neighborhood is an organic food co op, coffee shops, nifty looking hair salons, laundromats, boutiques, gastro pubs, a record store that rivals all others & the best independant dvd rental place in all of asheville. Most of these great places to walk to have just opened with in the past few years & I'm pretty sure more will come. I guess you could call it an up & coming neighborhood.
--I now live in woodfin/north asheville(just north of unca) & although its really safe, its pretty dull & I haven't even talked to some of my neighbors even though I've lived here for 3 years. There are pretty houses & I'm sure some streets are more lively(in a good way) than others. Being close to botanical gardens is nice & almost anything you could need is not a far distance away.
--Kenilworth neighborhood north of biltmore village is nice historical area but I have no personal experience with it. Its pretty & definitely not scary.
--I never really go to neighborhoods in east asheville but I've never heard of any areas to avoid over there. The public pool at the nature center is huge & you'd be closer to one of the best swimming holes out that way. East asheville is also very convenient to an entrance to the blue ridge parkway which is amazing.