From current fellows: The near east side (the isthmus and the area just east of it), which is a funky, liberal neighborhood with lots of shops, restaurants, coffee shops, bikes, dogs, babies and tattoos. It’s near a lot of parks, has some great bike paths, and gives access to the lakes pretty easily.
Downtown - near the Capitol Square is home to some of Madison’s best shopping, restaurants, and nightlife. It has the most “city” feel of any of the neighborhoods in Madison and is easily accessible to the hospital and all of what Madison has to offer!
While there are many others, here are some sites to get a feel:
Destination Madison - Neat Neighborhoods
The Five Most Popular Neighborhoods for Madison Renters