Угораздило has slightly more noticeable connotation of some involuntary event, of something, that happened agains the will of the subject, like in phrase "Эк меня угораздило въехать в эту стену". Not necessarily one managed to do anything to and up in such situation :)
Ухитриться and умудриться are very closed synonyms but not not absolutely identical. Yet both ухитриться and умудриться are about being able to achieve something that was hardly expected to be achieved - or was not planned to be achieved at all, you can say "ну и я как-то ухитрился всё-таки подобрать код для входа в подъезд", but you can not say "ну и я умудрился как-то всё подобрать всё-таки код" - it will sound slightly off.
"Ухитрился" is still applicable for voluntary actions. One can intentionally try to do something and succeed. "Умудрился" is more about ending up in situation one were not willing to end up with at all. So how does it differ to "угораздило" then?
Well, угораздило is more about unpleasant situation and умудрился is slightly more about unlikely situation. Угораздило попасть в пробку - is more like you didn't want to be in a traffic jam at all but well, shit happens. Умудрился попасть в пробку, - is more about like you never expected to stuck in traffic jam in this time of day, but somehow, you've managed.
Here's small plot to give you an idea:
All that said, all three words are still very closed synonyms.
You gotta get used for them in everyday situations to learn when they can be used completely interchangeably and when one is tiny bit more preferable.
