Housing in Madison is very versatile and offers different options. Before searching for your future apartment in Madison, one thing to keep in mind for those who are moving to the area is that the main tenants who are renting in Madison are students (because UW-Madison makes up a large part of the apartment rental economy) so leasing period follows the academic year. The majority of apartment leases start in August and end the following August which means that your options for apartment rentals are going to be limited in the Spring as opposed to the Fall.
First, the University of Wisconsin-Madison has a dedicated housing services open to students, faculty and staff with various units (1, 2 and 3 bedrooms) are available for rent at the University Apartments (see link below). The Eagle Height Apartments and University Houses give priority to students/postdocs with families while the Harvey Street Apartments are mostly reserved for single graduate and professional students. Eagle Heights and University Houses offer a community atmosphere for postdocs/graduate students who have young children since there is a daycare center as well as various cultural events that are planned year round. Available units are rare and quickly rented. Be sure to put your name on the waiting list as soon as you know you will be relocating! Check out the site below for more information.
University Apartments: https://www.housing.wisc.edu/universityapartments
The second option available is to use Craigslist, the online classified website: http://madison.craigslist.org. An additional website, which allows easy apartment searches, is Madison Apartment Living: http://www.madisonapartmentliving.com/
There are 121 registered neighborhood associations in Madison, most with high livability scores (see for example: http://www.areavibes.com/madison-wi/neighborhoods/). The neighborhoods with the highest number of postdoctoral students include Marquette, Tenney-Lapham and Eagle Heights, but other neighborhoods, including Bay Creek, Atwood and Regent-Monroe offer both ease of access to the University and a good number of local services.
Finally, as it is common elsewhere, apartment complexes own by large developments are less flexible in term of leasing contracts as compared to small complexes own by one landlord. This is an important consideration as postdoc training time frame is more fluid.Other recommendations from current and former postdocs: