I've used Led strips (basically surface mount LED's on a flexible base) in my cargo van - great illumination vs. power draw. Can't remember the brand - Wally World had them as a multi-pack on sale, so I thought I'd give 'em a try. I went with white LEDs, they were originally marketed to the "bling" crowd, so they do come in other colours too.
1156 conversion LED bulbs are coming down in price, if you go with sockets for these, you can always put in yellow ones for bug season.
If you aren't using the lights that much, you won't need a huge solar panel - start with something like a 15watt panel and you can add panels if needed. My memory is not good enough to reliably remember to manually charge a battery in an out of the way place like your shed.
LEDs will take care of "#*!;?%#@ it, just tripped over something again", and have good output vs. power drawn. If you have an occasional need for high output task lighting in the shed, I'd wire in additional 120vac lights and run a generator as needed.
If you want to try the battery+inverter+120VAC light route, be aware that some (many?) CFL bulbs don't play well (ie. die) when running off an inverter. If I get back to experimenting with this combo myself, next up I'll try dimmable CFLs, as I suspect they'll deal with/survive the switching activity of the inverter better than standard CFLs.
Rgds, D.