When Thomas Thorpe published Shake-speares Sonnets [sic!] in 1609, the sonnets (or at least a subset of them) had already circulated in manuscript for some time. Francis Meres already mentioned these circulating manuscripts in his Palladis Tamia, published in 1598.
Some of the sonnets had previously been published by William Jaggard in 1599 in the anthology The Passionate Pilgrim. However, of the 20 poems in this anthology, only five are considered authentically Shakespearean.
We don't know with certainty what sort of manuscript Thorpe used for the 1609 edition of the sonnets, but scholars now regard him as a reputable publisher. William Jaggard is less reputable, both because of The Passionate Pilgrim and the False Folio of 1619. His access to manuscript of authentic Shakespeare sonnets must have been limited.
Since the sonnets had circulated privately in manuscript form, it is worth asking whether any such manuscripts have survived. If yes, how many of them? And were any sonnets more popular than others?
Note: The question title intentionally reads "manuscripts of Shakespeare sonnets" instead of "manuscripts of Shakespeare's sonnets", because such manuscripts may have been incomplete.