I had the same thing happen as a very new student. The instructor flatly refused to fly, despite my pleas - I didn't know any better at the time - that would it really matter, it was a nice sunny day, etc. A good learning experience for me.
I also had alternator failure when on a dual cross country with an instructor. We just turned off all necessary electrics and flew home, no hassle.
We had a spider in the pitot tube recently. We found it before flight, and and went no further than to maintenance. Now I'm always super-careful to check my ASI is working before takeoff; sometimes I even taxi a little fast to check it, if I have the opportunity.
The good thing about these sort of incidents, especially if you've got someone more experienced with you, or they happen on the ground, is that you never ever forget what to do if they happen, or what to check to make sure they don't. And you stop believing that it won't happen to you!