As a "pure" technique, phase is quite hard to measure and needs more complex receivers - so it costs more than AM or FM for no benefit so it wasn"t common.
However, phase is used almost universally for anything more complex than a traditional audio broadcast. In particular, all modern communication systems (WiFi, 3G, satellite radio, digital TV) do use both phase and amplitude to transmit data.This is Quadrature amplitude modulation with quadaruture meaning 2D, with signals representated by complex numbers and diagrams like this:
Each dot (state) can be written as X+jY or as amplitude and phase.
Phase noise (how well you can resolve that) is a critical and hard thing in designing modern radios.
Frequency Modulation: Why is there no phase modulation radio?