How to trace a bitmap into vector shapes for free
Vector > Vectorise Bitmap in Paint by Zaweba traces a picture into editable shape layers with a live preview, and can write the result as SVG.

A bitmap can be traced into vector shapes in Paint by Zaweba with Vector > Vectorise Bitmap. A built-in tracer turns the picture's colours into shape layers that you can recolour and reshape, and it can write the result as SVG as well. Settings for the number of colours, smoothing, corner angle and minimum area, with a live preview, let you balance detail against clean shapes.
Tracing is the usual answer when you have a logo only as a small JPEG, a scanned drawing, or a piece of clip art you want to print large. Pixels go blocky when you enlarge them; vector shapes stay sharp at any size.
Choosing the settings
Colours sets how many separate colours the tracer looks for. A two colour logo wants two or three; a flat illustration may want eight or ten. Too many and you get dozens of tiny shapes for every shade of anti-aliasing; too few and parts of the design merge.
Smoothing evens out the jagged edges that come from pixels, giving cleaner curves. Corner angle decides how sharp a bend has to be before the tracer keeps it as a corner rather than rounding it. Text and geometric logos want corners kept, so keep that setting tight; hand drawn artwork usually looks better softer. Minimum area drops specks smaller than a given size, which cleans up dust on a scan or noise in a JPEG.
Two more choices shape the result. Stacked builds the picture from layers sitting on top of each other, which is easy to edit; cut-out makes each shape with holes where others sit, so no shape hides beneath another. Drop background removes the background colour so you're left with just the artwork.
Working within its limits
The tracer works on pictures up to 1600 pixels, with a 480 pixel preview. That's plenty for logos and icons. For a large scan, crop to the part you need first, or scale it down, because fine detail beyond that won't make a better trace anyway. The best source is a clean, high contrast image: if you can, tidy the picture with Levels to make the background white and the lines solid before tracing.
After the trace
The result is ordinary shape layers, so all the vector tools work on it. Recolour a shape by changing its fill. Use the boolean commands under the Vector menu, such as Unite and Subtract Front, to merge pieces that were split or to cut a hole. Smooth out a bump by editing the path with the Pen tools. Align and Distribute help tidy up parts that should line up.
When you're happy, File > Export writes the shapes to SVG with their fills and strokes, ready for a website, a vinyl cutter or a layout program. Keep the working file as a .ptx too, so the layers stay editable.
Where tracing falls short
Photographs don't trace well. A portrait has thousands of subtle shades, and a trace reduces it to flat patches, which can be a nice poster effect but isn't a faithful copy. Tiny text in a logo may also come out lumpy; it is often quicker to retype it with the Type tools in a matching font and convert that to shapes.
Paint by Zaweba is free software and traces on your own computer, offline, so client logos and private drawings never get uploaded anywhere.