I had this feeling twice now in 3 months so it's not that often. It hurts though.
Sitting at work behind my desk and suddenly a pain starts in my shoulder and travels, feels like a line, over my chest to my other shoulder. Didn't do anything weird or sat in a weird position. The pain stays for like 10 seconds and then subsides. Any clues?

Anything from 'nothing' (muscle strain from how you sit at your desk, usually at a computer, where you are raising one shoulder a tiny bit to use a mouse) all the way to 'something' (perhaps heart related).

Symptoms are generally useless without the whole person. We know nothing about you, nor your history, nor your family history, where you live/age/weight/gender/diet/pills/supplements/cigarettes/alcohol do you get the picture?

That's why you have a doctor or a regular clinic where your records are.