We're really getting into Malcolm Douglas territory here, but I think it's a safe bet that some of the songs were created by Shakespeare (assuming he also created the plays, which for my money he did); some he borrowed, and some were put in later.

Lear, Hamlet and Othello are the best endowed with song, and some of the verses in these are thought to be original. However snatches of folksong are ascribed to Ophelia and Edgar as a means of indicating madness (which must prove something!). And I believe Desdemona's Willow Song was a known song of the period, with words and music still extant.

I believe the Pyramus/Tisbe interlude in Midsummer Night's Dream is a parody of a genre that had been around for many years, and "Oh mistress mine" in Twelfth Night predates the play I think. Sometimes a song is intentionally borrowed, as with the drinking song in Antony & Cleopatra, "Come though monarch of the vine...." which was well known before the play.

The norm for songs in Elizabethan plays was that they would be sung to pre-existing well-known tunes. Whether Shakespeare ever indicated such tunes I don't know.

Part of the difficulty is that the publication of Elizabethan plays, Shakespeare's included, was not based on anthors' texts but on performance. Thus the first Quarto edition (1611? I may be wrong on that) of Hamlet was sourced from actors' memories and prompt notes. By the time it appeared in the first "collected works" - the Folio edition of 1623 - there were significant differences in structure and text. Or sometimes the text would be similar, but ascribed to different characters (eg the gravedigging scene).

On top of all this, the Puritans then gained the whip-hand in England, and entertainment of every sort was knocked on the head. When Dryden and others later took advantage of the Restoration to rescue Shakespeare, they also took ever increasing liberties with the known texts, and even the plots. This trend culminated in Nathaniel Tait's Lear, from which the Fool, now regarded as one of the four main characters - was excised altogether. The trend was only reversed by scholarship in the 19th and 20th centuries.