The most popular instrument at this period is the voice, so one way to gauge the popularity of a tune in this age, at a time when there was no radio or recording technology, is by how often it was used as the tune for a newly-composed song or ballad. After that, there's lute and keyboard instruments like virginals, harpsichords...so a popular tune would become a theme for some a piece of theme-and-variations for those, also. For both categories, you would certainly find Greensleeves ( yeah, even then it was an ear worm, though often played fast, like a jig). Packington's Pound, Lord Willoughby's Welcome Home, and Go From My Window are others. The 17th c. English would develop a taste for melancholy tunes in 6/8, with a dotted rhythm...yeah, like Greensleeves.

There's also theater songs, of course. Robert Johnson the lutenist famously did a lot of music for Shakespeare's plays, and you'll find transcriptions of them published, arranged for recorders, guitar...played on YouTube. You could browse the [list](https://en.wikipedia.org/wiki/Robert_Johnson_(English_composer) at the bottom of this article for the names, search for them online and you'll likely find still lots of people playing them and singing them.. Yes, it's a Wikipedia article, but I only said browse the list of works, don't depend on the article as a source for his life