All of the tutorials I’ve seen for this seem to make things harder than they need to be. Hopefully this helps a few people!
This is what I do to cut out an image or remove a background with Paint.net:
Open the image you want to work with in Paint.net.
Add a second layer in the Layers box (Ctrl >> Shift >> N) and and make sure it is highlighted in blue so that you’re working with it (Paint automatically names this Layer 2).
Using the Line/Curve Tool, trace around the object you want until you are back where you started and the line connects to itself. It helps to zoom in when going around detailed areas. If the object you want goes to the edge of the original image, just draw a straight line across, staying close to the edge of the image. I use a bright blue or red color to make it easier to see. You can also click the little circles while using the Line Tool to make curves. Make sure each line you start connects to the end of the one before so there are no gaps.
Working in the same layer as your traced line (Layer 2), use the Magic Wand Tool and click the object you want to keep so that it is highlighted (the part inside the outline you just drew).
Now go to your Layers Box and click on the FIRST layer (called Background). Do NOT click anywhere on the image yet. You should now be working in the Background Layer and your object should still be highlighted.
Click Ctrl-X (Edit >> Cut) to cut out the object you want, with none of the traced lines from the layer above.
Now open a new Paint doc with Ctrl-N (File >> New).
Add a new Layer to the new doc (Layer 2) and make sure it is highlighted in blue.
Click Ctrl-V to paste your object onto Layer 2 (the transparent layer).
Uncheck or delete the white Background Layer. The object should be floating on a checkerboard background and if you save it as a PNG it will flatten the layers and retain the transparent background.
Have fun using your new object in your projects! 🙂