Community Wiki: Which public transport networks allow you to transport a bike?
New England

Massachusetts

The MBTA in the Boston area has bike racks on a fairly substantial number of the busses (70%, according to the web page). The Red, Blue, and Orange line subways, and Commuter Rail allow bikes outside of peak hours.

The Pioneer Valley Transit Authority in Western Massachusetts (Amherst/Northampton area, with several colleges) has bike racks on their busses.

New Hampshire

Manchester Transit has bike racks on most busses. Some busses have bike racks which accommodate three bikes.

Concord Area Transit has bike racks on most busses.

Nashua Transit has bike racks on all busses.

Advance Transit in the Upper Valley region of Vermont and New Hampshire (Hanover, NH, White River Junction, VT, the Dartmouth College area) has bike racks on many of their busses.

Vermont

CCTA (Burlington area) has bike racks on all busses, as well as bike lockers. 

GMTA (serves rural Vermont, and recently merged with CCTA) also has bike racks on all buses.

The Amtrak Vermonter rail line (connecting St. Albans, Essex Jct. (Burlington), Springfield Mass., New York City, and Washington DC) now offers carry-on bike service as of May 2016. (There is a limit of 3 bicycles per train, a $10-$30 fee depending on destination, and bicyclists are asked to arrive early at the station)

Maine

METRO Bus (Portland) has bike racks on all busses.



Even busses with bike racks can only carry two bikes at once, which means that if many people use them, you may need to wait a while before finding a bus with space, and some of them have restrictions (such as not on express busses, or not when the busses are replacing a train). In general, I find it's best not to rely on being able to take bikes on public transit, but use it as a backup option for when the weather turns bad, you have an injury, or something of the sort, and you still want to take your bike home.