Cut shape in Inkscape so it has transparent background?

I was wondering if Inkscape has the ability to select the outline of a shape, and then be cropped or copied and pasted over another shape?

I remember being able to do this in Illustrator but you can't seem to do it in Inkscape.

Setting a clip for an object keeps coming up (in searches) but that doesn't seem to do anything. If you paste it just pastes the object with a rectangle still around it. All I want is to select a shape and then paste it with a transparent background. Can Inkscape do this?

Any help much appreciated

Rowan