MME (Durham Tees Valley) has an odd rail service. It does have a station, bearing the old name of the airport (Teesside Airport). However... Despite regular trains on that line, only one per week actually stops there.

Otherwise, in the UK...

LHR (London Underground, Heathrow Express, Heathrow Connect)
LGW (Gatwick Express, Southern, First Capital Connect, First Great Western, Virgin)
LCY (Docklands Light Railway)
STN (Stansted Express, One, Central Trains)
BHX (Central Trains, Virgin - connected by a light rail service)
MAN (Northern Rail, Trans-Pennine Express)
NCL (Tyne and Wear Metro)
SOU (South West Trains)
CWL (Arriva Trains Wales)
PIK (Scotrail)

LTN has a dedicated station, but it is not on-airport and requires a bus link.