I see where the normal range of the clarinet is low E to high C above the staff. I know the clarinet will go higher than the normal range but is it common to come across music where the clarinet must play above the normal range?

It is quite common for the clarinet to go up to G (& even beyond sometimes) especially in 20th & 21st century music.

I would extend the practical, normal range of the clarinet from E below the staff to the G in the 4th ledger line above the staff. Up to that G is quite common in solo and ensemble literature. I have seen up to a D above that personally, but only in modern solo works. The highest note I have seen in a standard piece of rep is the high Bb in the first movement of Weber's second Concerto.

The Artie Shaw Concerto has a gliss at the end from G to high C. Loads of fun.

If this is a question originating in the idea that maybe "I don't have to learn all of the notes if they aren't used very often" then forget it. If you don't learn all the fingerings, you will regret it later.
If not, then that is a very intriguing question. I've seen clarinet music go quite high in many pieces, but I've never actually thought about how high "normal" is.

I wouldn't consider there to even be a normal range. All the music I compose goes all out to the extreme on both low and high ranges. This appears to be very common in Classical music, and 21st century music. Mozart's Clarinet Concerto (bother versions) go all the way to either extreme. It shouldn't take you more then a day or two to learn those few extra fingerings. I learned all the fingerings on my first day, whether ot not I had the embouchure to play them was questionable, but 2 years later, I could play the entire possible range easily. I have been playing sense I was 4.

Correct me if I'm wrong, but I believe the Mozart only goes to a G, which I would not call the "extreme" of the range...