Can mechanical waves travel through matter and through empty space?

Electromagnetic waves.

Mechanical waves (like sound) require things to bang into each other, or at least push each other around, to continue to be a wave propagation. In space, there isn't enough stuff close enough...

Electromagnetic radiation, such as light, infra-red, UV, X rays.

A light wave can travel through space. Otherwise, how would sunlight reach the Earth? Also, light does have wave-like structures, and it is made up of particles called photons.

You could have Gamma, ultra violate, infrared and radio