Yes, you are permitted to bring a purse or small bag on the vast majority of rides at both Disney World and Disneyland. Unlike some other theme parks that require you to use lockers, Disney designs its ride vehicles to accommodate small personal items. On most attractions, you can simply keep your purse on your lap or place it in the mesh storage pouch located in front of your seat. For more thrill-focused rides like Space Mountain or Expedition Everest, it is recommended to wrap the strap of your purse around your leg or arm to ensure it doesn't slide around. The only exceptions are very few attractions where a Cast Member might direct you to leave large bags in a designated area, but for a standard-sized purse, you can almost always keep it with you throughout the experience.