This question is similar to an earlier one about famous actors. It was very difficult for anyone to acquire "fame" in a era when there were no newspapers, no magazines, no television, and not even any "public" concerts (concerts to which the general population might purchase a ticket and attend). If and when a singer was "famous," that fame was largely limited to only the highest socio-economic classes that could afford to hire them for what were effectively private concerts. "Popular" music was mostly madrigals and folk songs, and the writers or composers of those tunes were largely anonymous. Opera was not yet in existence (first opera was in 1598). Celebrations to mark various occasions (example: weddings of royalty and the upper nobility, entertaining important visitors) were often accompanied by "masques," or short stage plays with musical accompaniment, but the participants were quite often the celebrants themselves. Masques were, by definition, amateur performances. Rich people liked to dress up and perform, and most educated people were taught music. The general public was not invited to those events. There were traveling minstrel shows, and some of the singers in those minstrel shows might be well known to his/her specific audience, but they were not "famous" in the modern sense. Virtually no singer of the era would have been recognized or celebrated by the general public. There were no Ed Sheerans or Beyonces, just good singers who were heard once or twice and quickly forgotten.
I should add that "fame" did not become "a thing" for either singers or actors until a century or more after the Tudor period. And the first musician or singer to become a "celebrity" in the modern sense did not appear until the early 1800s (pianist Franz Liszt).
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