"The Gods of Love that sits above", Much Ado About Nothing: this is the opening of an old ballad by William Elderton. Tune is on my website, B163. Song is in the Braye Manuscript at Yale, and the administrater of the SCA Minstrel website is contemplating getting a microfilm copy of the manuscript to get the whole song. The first verse is on the SCA Minstel website.
"It was a lover and his lass" in As You Like: It has a tune by Robert Morley, 'The First Book of Ayres', 1600.
"O Mistress Mine", Twelfth Night: printed and MS copies are noted in Seng book cited above.
"Farewell deere heart", Twelfth Night: Music and a version of the song in Robert Jones' The First book of Songs and Ayres, 1600.
"Hey Jolly Robin", Twelfth Night: Tune by Wm Cornyshe c 1520. Seng gives a complete copy of the song.
"In youth when I did love", Hamlet: Vaux's "I loathe that I did love", Tune is B216 on my website
"Take o take those lips away"; Measure for measure: tune composed later, c 1650 by John Wilson.
"The poore Soule sat singing, by a Sicamore tree", Othello: Probably not by Shakespeare. Broadside version of the song and its contemporary tune in Scarce Songs 2 on my website.
"Hearke, hearke, the Larke at Heauens gate sings", Cymbeline: See Sengs for tune in manuscripts.
"Jog-on, jog-on, the foot-path way", The Winter's Tale: Tune B251 and B252 on my website, but did Shakespeare write the song? Longer copies are known.
"Full fadom fiue thy Father lies", The Tempest: For an early tune by Robert Johnson see Seng.
"Where the Bee sucks, there suck I", The Tempest: for early MS copies of the tune see Seng.

In addition, there are several recognizable fragments of old ballads in Shakespeare's plays, (e.g., "Some men for sudden joys do weep, but I for sorrow sing"- John Carelesse) but the original tunes are unknown. There's also a tune used in Hamlet for Ophelia's song from the early 18th century, but it is not known how far back this goes, (Tomorrow it is St. Valentine's Day, on my website).