i found out (and i write it here hoping it can be an useful piece of information) that if you have enabled the 2-steps verification on your microsoft live account, that is, the more secure login where they also ask you for an additional code you get from your phone, then you cant login into messenger.
This is true, unless you have enabled, on windows 8/8.1/10, the online microsoft account. In this second case, logging into your pc grants somehow the entrance to messenger.
I noticed it because i recently disabled the online account i was using on my main computer, and reverted back to the standard offline account, after this i wasnt able to login into messenger. In order to solve my problem i entered into my microsoft account from the browser, i turned off the 2-steps verification, and… problem solved!