When I asked my parents (born in rural China) about their youth, one thing that struck me is the lack of motorcycles. Those who could get assigned motorised transport i.e. government officials, all seem to have gotten cars. This led to me wondering, did the Chinese public (non-military) even use motorcycles before 1976? If so, how would you qualify for eligibility to have a motorcycle? What kind of motorcycle would you usually get?
Yes there were. The first motorcycle entered China in 1886, during the Qing dynasty. It was imported into Shanghai, which at that time was partially occupied by foreign countries, and so had strong links with Europe.
But I assume you're talking about the PRC. The first motorcycle was produced in the PRC in 1951, under the Jinggang Mountain brand (井冈山) and were based on the German Zündapp K500. They were named after the location of the birth of the Red Army, the precursor to the People's Liberation Army. From then, various brands were developed, all in imitation of foreign models. Most were made for military use and many had side cars. By 1954 three-wheeled motorbikes were in production.
To be honest, it's hard to separate military and civilian ownership of certain things during this period. Yes, people had possessions, but the PRC was a communist country (or, to put it in the language of the time, they were marching down the road to socialism) and so private ownership was frowned upon. This was especially true for military equipment (although that's not a particularly commie trait).
As for who could use them, most likely if you didn't use one for work (in the military or otherwise) then you probably wouldn't use one at all. Most people got around on bicycles and didn't have the money for more expensive forms of transport. Initially, motorcycles were exclusively produced for the military, then also for public security organs (公检法), that is, for use by workers in the Public Security Bureau, the Prosecutors Office, or the courts.
Sources:
中国生产的第一部摩托车是什么品牌的？有何历史渊源？ This link has some photos.