What is the best thing to do, stay still, paddle away, or what?

it depends how close the shark is to you

the only reason the would ''attack'' is because they think you are something else. i.e. a seal.
some say to sit up on the board cross legged, and others say to get off the board and make sure you show yourself to the shark as human and not a snack. just dont splash around.

Use your board-mounted harpoon and get merry with the sushi!?!

Seriously though, unless the shark is really close, l would not worry about it. There are alot of sharks in the water that have no interest in eating people, even by accident. If it is really close do not splash around or panic, just stay calm and my advice would be to swim to the shore slowly and calmly.

Odds r in ur favor that the shark wo not bother you. l know of a guy who had a chunk of his board bitten off by a shark, paddled in, borrowed another board, & went right back out. A lot depends on whether baitfish r running & birds r floating on the surf, It also depends on how YOU feel, ur instinct. Myself, if l see a shark close to me, l paddle in. l also wait until the baitfish have finished running through. l know that Sharks r always in the water, but try to be cognizent of my surroundings. Check out this site on advice.
http://flmnh.ufl.edu/fish/sharks/Attacks/relariskreduce.htm

Read though it. It has a lot of good information on understanding Sharks.

l would slowly and non-panicly paddle in while trying not to splash the water.

The Surfboard Man
http://thesurfboardman.com/

l guess that depends upon A) how it is behaving & B) how big it is. If there r sharks actively chasing bait fish in the lineup it is probably a good idea to go in to avoid an accidental bump-and-bite scenario, in fact many times l will paddle in if there is a large school of bluefish or something similar coming through the lineup even if l do not see sharks. It is up to u on the size one. l kinda go with my gut feeling, do l say ''oh hey there is a shark l better keep an eye on it'' or do l say ''Holy Crp!'' If it is the second one l am heading for shore.

it depends on how close the shark is to you. if it isnt close it prob wont bother you so either just sit on your board and wait or continue surfing. but just dont splash around like a wounded fish or seal because it will get curious and come right up to you

You probably noticed that some of the other answerrs had a few qualifications to add to their , ''It depends.'' It does depend on a lot of things. First & foremost, realize that sharks r not just eating machines, & shark attacks occur THOUSANDS of times less frequently than getting struck by lightning. And, sharks r in every ocean of the world (and bull sharks in lots of fresh water estuaries), & u do not see the beach littered with the mangled bodies of shark bitten surfers, do you?

l could fill volumes with shark lore, l wo not do that, but l do recommend u check the link that Surfammo included for you. It is VERY informative. Be mindful of what Surfammo & Kim_Dude said about the bait fish (that is what the sharks really want to eat, not you). However, not all of the advice u got from some other folks is right.

To keep it simple, without over-analyzing all of the possible scenarios, l would just suggest that u carefully, calmly & deliberately paddle into to the beach.

Two things to keep in mind:

You do not have to be faster than the shark, u just have to be faster than the guy who is surfing wth you, and
it usually is not the shark that u see that eats you, it is the one that u did not see.

Now stop worrying about sharks & go surfing!

idk just dont splash alot it attracks them but they'r usually not going to attack

cut your leg and make tons of noise.

seriously, unless you live in a ''hot spot'' for that, dont sweat it. i'm more worried about the sea lice biting my nuts.