I have an irregular shaped picture I'd like to save, in a format which allows the irregular shape to be preserved. Which image formats should I look at, which will support the irregular shape so I don't have to do any manual processing when editing the image?
No matter the format, your image will always be rectangular, that is, the usual width X height format.
As above answer, I am assuming you want to preserve transparency, and for that you would need either gif or png format. Png has better quality than .gif and is supported in most if not all mainstream browsers and image applications.
If you mean an image with a transparent background then the most useful format would be png. This has lossless compression and can support an alpha channel.
Most (if not all) image editing programs support reading and writing png format files.