I'm writing this article to show people why PNG is the best format for screenshots and why not using JPG and BMP for these.

I'll start this article with a few myths and explain why they're incorrect.

PNGs look better, but JPG compresses better.
Almost correct buddy.
JPG compress better for PHOTOGRAPHIC images.
In most of non photographic images PNG will compress better than JPG.
JPG was meant to be an image format for photos.
But when I compared it in my computer it wasn't the same!
This is probably because you used a program that doesn't fully implement the PNG compression, and therefore will make bigger than average PNG files. MS Paint for example doesn't fully implement it.

PNGs compress better, but unlike BMPs they still lose data
Totally wrong. PNG uses a compression algorithm without dataloss.
BMPs from the other side don't compress AT ALL. They store the raw data of pixels in this format:
RGBRGBRGBRGB...
Also, BMP is not a web standard.
It makes no sense! How can PNGs compress without data loss?
PNGs are very similar to ZIP files.
For example, when you put a text document inside a zip file, it will be a smaller file (Unless the text is too short!) but it will not lose a word suddenly!

My game only has 256 colors, it's a better idea to use a GIF file!
Partly right.
In 90% of the times a PNG file will be smaller than a GIF file.
No it's not! I check it myself and it's still bigger!
This is because you compare a PNG file with 24 bits color with a 8 bits GIF file!
By default on many graphics program PNGs are saved as PNG24.
You can save the PNG as PNG8 with indexed color by using a more advanced program than MSPaint, such as Photoshop or Paintshop.

Now for the second part of the article, I will compare the 3 formats (Without Gif, because it's limited to 256 colors) in a table.

Format |Size |Data loss |Preview
PNG |32 KB |No |http://clickteam.info/neatwares/stuff/LIJI/Formats%20Compare/PNG.png
JPG(Q0) |43 KB |Major |http://clickteam.info/neatwares/stuff/LIJI/Formats%20Compare/JPG-Q0.jpg
JPG(Q4) |56 KB |Major |http://clickteam.info/neatwares/stuff/LIJI/Formats%20Compare/JPG-Q4.jpg
JPG(Q8) |91 KB |Less Major |http://clickteam.info/neatwares/stuff/LIJI/Formats%20Compare/JPG-Q8.jpg
JPG(Q12)|184 KB |Minor |http://clickteam.info/neatwares/stuff/LIJI/Formats%20Compare/JPG-Q12.jpg
BMP |~750 KB|No |Looks the same as PNG

I hope you learned something from this article, and will start using the suitable format next time.