Load your image into the program

If you already own a different vector illustration program, this technique will probably work just as well in that program also, although the shape-drawing tools may work slightly differently.

I'm using DrawPlus SE Pro, the upgrade for SE for about £10 which adds more functionality.
However, I also tested it using plain DrawPlus SE (the freeware version from http://www.freeserifsoftware.com/ ), and it works just as well for this technique.

To load a bitmap image into DrawPlus, do not use the startup wizard, and close the default blank document.
From the File menu select Open to select and open your scanned image.

Draw a vector shape as a "mask"

Use Ctrl+scrollwheel on your mouse to zoom in and out on the area you wish to select. Click and hold the scrollwheel on the mouse to drag the zoomed image around behind the viewing area.

Draw a vector shape over the top of the area of your image you want to become a cut-out piece for your cut-out character.
In DrawPlus, simply choose the Pen Tool, and click, click, click like the reverse of a connect-the-dots puzzle - you are drawing the dots, the software will connect them. By default DrawPlus will change to using smart nodes for the connecting line, which gives a nice natural joining curve through the points you select. Don't worry if the fit isn't exact, we'll tweak that later. Make sure to just use a simple click-and-release, not click-and-drag.
(In Xara Xtreme 4.0, drawing with the Shape Editor tool is a little similar, but I find it isn't as smart with corner sharpness).

Click back on your first dot to finish and close the shape.

Make the mask shape mostly transparent

To see through the shape, make the fill mostly transparent. In DrawPlus, (1) Click the fill square, (2) select a nice contrasting colour from the colour picker (contrasting with the colour/s you are masking), then (3) choose a low opacity

Make the line (stroke) completely transparent

For more accurate alignment of the edges, turn off any line or stroke around the vector shape.
In DrawPlus, (1) Click the line rectangle, then click the small No Fill square.

Tweak the shape as needed

Using the Node Tool you can drag points around or even just grab and drag the line itself.

Drag the points to move the line to the outer edges of the shape you are masking

Use the mask to "clip" out the piece you want.

Select both the vector mask piece and the bitmap drawing. If you have been following this tutorial closely they are probably the only two objects on the screen, and you can simple you use the keyboard shortcut Ctrl+A (for Select All).

in DrawPlus SE you can then go to the Arrange menu, select the Crop sub-menu, then select either Crop to Top Object or Clip to Top Object
If you are not using DrawPlus, the precise location of this function will vary in your software.

Export the cropped object

From the File menu, select the Export sub-menu, then select Export as Image...

Some export settings

Choose Selected Items from the Export Area choices, Portable Network Graphics as the Format, and either 32 as the Bit Depth or 24 and make sure Transparency is selected (which is the same thing as 32-bit).
You should see the background shown as a light grey checkerboard background, indicating transparency. Click the Export button.

(If you adjust the size or DPI, you will need to do the same for all exported objects to keep them in proportion to one another.)

Move on to the next piece

After you have exported your shape to an image file (e.g. left-upper-arm.png), then you will be returned to your drawing.

Optionally, you can save your masked piece as a DrawPlus file (e.g. left-upper-arm.dpp) using File, Save As. Do not close the file yet.

Use Undo (Select Undo from the Edit menu, or use the keyboard shortcut Ctrl+Z, or use the Undo button at the top of the screen)

You can then delete the masking shape (in DrawPlus first use the Pointer Tool to select the shape, or you might still have both the shape and the image both selected and accidentally delete both).

You can now draw another masking shape over the next piece of your cut-out character.