Michael, any bolt action will fire all four types, but whether or not the shorts and BB caps will feed from the magazine is another question. You may need to check with the manufacturer to be sure before you buy.

My 1966 Remington 511 scoremaster 22lr Bolt action fires All as well BUT, The shorts DiD have a feeding issue from the magazine.

Michael, any bolt action will safely fire all four 22 cartridge's. Since you're looking for a bolt action feeding issues really shouldn't be a problem. only time I've had feeding issues with 22short is in lever actions and semi automatics. Here's a 22lr bolt action i'd recommend......... budsgunshop.com/product_info.php/products_id/712022899/cz+457+american+sr+.22+lr+20tb+1+2x20+walnut+5+round........... Hope this helps.

Like others said, fire? All. Cycle? None, at least with a box magazine. Tubular magazines are generally the only solution if you don't want to load single shots. The Marlin XT-22TR is a tubular magazine bolt action that would probably fit the bill if you can find one floating around somewhere, else there are several older Remington and Winchester models that would be on the used market. In terms of things that are still made presently, pump or lever actions from Henry, Rossi, Browning, etc. would be easier to find though not bolt action.

Michael, no worries on the clarification. Any Bolt Action Rifle chambered for .22 Long Rifle will also fire .22 Longs, Shorts, and even CB and BB caps. All of these additional cartridges have lower chamber pressures and are not unsafe in that regard. However, some types such as many semi-automatics and assorted others may have to have each round loaded individually and also directly into the chamber as the weapon may not reliably feed or eject these shorter case lengths. As such, the rifle will function strictly as a single shot weapon. Cheers and stay safe!