I see a lot of people mention using tape to hold the paper from spilling off the roll, or after rolling the paper up to hold it from unrolling. That works, but it usually leaves marks where the tape is placed and can tear the paper when the tape is pulled off if the tape has been left on a while.
My solution is to use plastic work clamps. Menard's, Lowe's, and many other home improvement places have kits of plastic spring-loaded clamps for under $10. In fact, I see that Menard's has them on sale this week. You get about a dozen plastic spring-loaded clamps of several different sizes for $5.88.
After unreeling the paper stick a clamp on the ends of the roll and it won't unspool more paper. Roll the paper up and stick clamps in both ends and it won't unroll. Clamps are easier for me to deal with.
And the clamps have a lot of other uses...