My go-to fastener is a coarse-thread drywall screw. Good grip, strong for it's size, V-bottom head for countersinking, and you can get them in most any size you want.

Past that...I learned long ago that each screw, nail, bolt, nut, and washer I can save has a good chance of keeping me from having to run to the hardware store. The trick is to keep it all ONLY semi-organized...and for me, it works out best to split (ex: screws) into three groups - small (<3/4") medium (<2") and large (2"+). When I need a medium screw, I dump some of the container into a bowl I keep on hand for it, and parse through the available screws to find the size and type I need. If I don't find what I'm after, other options are still right in front of me that way.