The GT-1000 running shoe from Asics is a good all-purpose running shoe for runners of all ilk. Here are the basics if you're considering trying them out:
Support: Dual-density midsoles (the part directly beneath the inserts that absorbs shock) and a firm medial post (underneath your arch) help with stability issues.
Comfort: Less padding than their predecessor, the GT-2100 series, but gel inserts and lightweight mesh uppersoles make for a pretty soft run.
Durability: The GT-1000's last for awhile, but I personally have cut through the mesh at the toe with every pair I've had.
Style: Not flashy neon statement shoes, so they go with pretty much anything.
Price: Mid-range, but at $100, the additional features for stability and comfort mean you're getting your money's worth.
The Basics: I used to run with the GT-2100 series, but after Asics ended that line, I switched to the GT-1000 and have been pretty satisfied since. I tend to overpronate, and these (plus my orthotics) help correct that. Check out the GT-1000 at the Asics website, and get a bit more technical with a review from REI (where they're on sale for 25% off!) and from Runner's World, where the GT-1000 won their "Best Buy" award in 2012.