What do Electromagnetic waves actually look like in space?

Well they don't really have a specific shape because they oscillate in time. But a single wavefront is just a spherical pulse that radiates outward, and at large distance from the source can be approximated as a plane wave.

 

Correct, the sinusoidal component of an EM represents the strength and polarisation of the EM field, not physical displacement as in the case of an acoustic wave.

Claude.

 

The 'wiggly line' you see in representations of EM waves is not a picture of how something 'looks'. It's just a graphical representation of the amplitude and direction (vector) of the Electric and Magnetic Fields.
A similar thing can be done when plotting sound waves - which are longitudinal, of course. You can represent the instantaneous pressure at a point in space on an axis at right angles to the movement and this will also look like a wiggly line. It is not a picture of what happens but it is is a lot better way of showing what goes on than trying to draw dots and lines of different spacing, which is very difficult to do, for a start!