square wave carrier signal
When sending data over a cable or as light pulses, modulated square waves are common since it's easier to produce square waves than sine waves.  When sending radio waves over the air, however, there are generally very restrictive rules about transmitted spectral content; because of those rules it is generally desirable to avoid using much bandwidth than would be necessary to send information.

If there were no other radio devices in use and one was allowed to transmit whatever and however one wanted, it's possible that sending a modulated square wave might allow better performance than would be achievable with the same level of power using a comparable amount of circuitry; many kinds or resonant structures inherently behave like a comb filters, which will very nicely capture all the harmonics of a square wave just as effectively as the fundamental.

Nowadays, however, one isn't allowed to transmit whatever and however one pleases, but is instead required to confine transmissions to particular frequency bands.  Amplitude-modulating 300-3300Hz audio content on a 1MHz square wave would generate spectral content from 996,700Hz to 999,700Hz; 1,000,300Hz to 1,003,300Hz; 2,996,700Hz to 2,999,700Hz; 3,000,300Hz to 3,003,300Hz; 4,996,700Hz to 4,999,700Hz; 5,000,300Hz to 5,003,300Hz; etc.  The higher-frequency spectral content might help slightly with one's ability to receive the signal, but the potential improvement in receiver performance would be nowhere near sufficient to justify the huge increase in spectrum usage.