The most important feature of any folding bookshelf is stability. Regardless of what style of folding bookshelf you choose, or how small and compact it folds up, the shelf will be useless if it is not stable enough to hold books without falling over or collapsing. Be sure the overall structure is sound and able to hold the weight of several books. Once you have determined the strength and durability of the folding bookshelf, it is okay to consider its folded size and the ease with which it can be folded and unfolded.
Carefully consider the materials used to construct the folding bookshelf, including any joints or hinges. Wood, plastic, and metal are common materials for the shelves and frames of folding bookshelves, and each material has its own set of advantages and disadvantages. Wood, for example, is perhaps the most aesthetically pleasing material, though it may not fold up as compact as other shelves and it can be prone to water damage, chipping, or other damage depending on the type of wood used. Metal is not as visually appealing in most cases, but it is often lightweight and it folds up very compact. Plastic shelves tend to be flimsier than other materials, but it is usually the cheapest option, the lightest weight option, and the easiest to fold up very compact.
Most folding bookshelf models will feature some sort of hinge system, or some other connecting system that will allow you to either break down the shelf into different pieces or to fold the unit up into a compact, flat shape. Consider how the folding bookshelf folds and unfolds; if the shelf is hinged, examine the hinges carefully to make sure they will last a long time, will not break with repeated use, and will allow the overall structure to be stable when loaded with books and other objects. If the pieces break down and are connected with fittings or slots, make sure the system is easy to use and stable when set up.
Aesthetics is, of course, an important part of your folding bookshelf, so be sure to choose one that will fit with the overall aesthetic of the room in which it will be placed. Inexpensive plastic shelves are great for dorm rooms and closets, while wood shelves are great for the living room, kitchen, or other room in which the shelf is likely to get noticed.