JPEG is most commonly used.
You won't be able to shoot in GIF or PNG - however, should you have the chance to convert for storage, consider TIF (or TIFF)... lossless, unlike JPEG. PNG is also lossless.
It just means that when you edit a lossy file, like JPEG, it will have consequences on the quality. But edit a PNG or TIFF and no problems there.
However, most cameras can only shoot in JPEG. Almost all DSLRs allow raw (or RAW, both are the same as it does not stand for anything) files and some allow TIFs.
Raw is the more preferred options for professionals - not PNG, unlike what people can claim. You shoot in raw (the file extension differs for each maker: Canon uses .cr or .cr2, Nikon .nef, etc. - and there is also one by Adobe, trying to unify all raw formats; .dng, or digital negative) and then develop it in a raw processor, such as Lightroom, Camera Raw, etc. and then export it as either JPEG or TIFF. An edited raw photo will not have changed if you edit and save it - to retain edited changes, you have to save as either a JPEG or TIFF, depending on whether you want to further edit or not. TIFFs are a lot larger in size than JPEGs, which are compressed.