This is primarily opinion based, but as someone who has taken a lot of calculus and engineering notes in LaTeX I feel like recommending Lilypond, which is basically the LaTeX of music:
The primary advantage of such a tool is that it can produce very beautiful scores of all kinds, including neumatic notation and stuff: http://www.lilypond.org/examples.html
Most importantly, if you are a touch typist you don't need to take your eyes off the blackboard, you don't need to fiddle with the mouse and go through menus instead of paying attention in class.
You can actually not only paste it into Evernote (since it's text based), but you can embed it in a LaTeX source like an equation, so you can have beautifully typeset notation inside beautifully typeset lecture notes: http://www.lilypond.org/doc/v2.17/Documentation/usage/latex
The downside is that - for the basic things the learning curve is not really steep, but you need to have a look at the manual first, you can't get away with guessing your ways through menus.
Personally, I think the advantages are enormous, but there is an army of people out there who just won't touch anything that's not strictly WYSYWYG and menu-driven, so if you are one of them, it might not be your cup of tea.
EDIT: There is a web version, apparently: http://weblily.net
EDIT2: There is also a GUI which may be useful for beginners: http://denemo.org/