Basically, you can run any width of tire that can safely clear the brakes and frame and which is not "too wide" for the rim.
When considering brake/frame clearance you need to understand that a "wider" tire will not only be wider but "taller", so make sure that the outer circumference of the tire will not rub against the frame or the brake pivot (in the case of caliper brakes). And if the new tires have a deeper tread than the old ones, account for that as well.
What's "too wide" for a rim is hard to say -- there are no rigid rules. My rims are about 20mm, yet I run 45mm tires (and this is the way it came from the manufacturer). Your rims also look to be about 20mm, so rim width is likely not an issue for you -- you could easily run 35mm tires on them (if you have the frame clearance).
When you install wider tires, be sure to carefully check the brake pads to assure they do not rub against the side of the tire at all, when brakes are being applied (since this will eventually wear a hole in the tire -- ask me how I know). It may be necessary to move the brake pads a little lower (or have a bike shop do it) to eliminate potential rubbing.
I'd suggest seeing if a 28 or 32 will fit.