Shakespeare did. A recent article in the local daily suggests that the conspiracy theorists are gaining ground, but read closely. The idea is that Shakespeare, a provincial yokel who never attended university or traveled outside of England, could not be the author of some thirty plays that do not so much depict the world as reproduce it on the page and stage in all its many colors. Writing is confessional; you have to have lived it to describe it; and the Earl of Oxford, not Shakespeare, has the needed biography.
But the Earl of Oxford died in 1604. Around ten of the plays were written after that. In a more fundamental way, the Earl of Oxford could not have lived them, or written them. William Shakespeare did.
The article names some of the Earl of Oxford's champions: a few actors, a newspaper guy, three justices of the Supreme Court. Call me an elitist, but why are there no literary scholars among them? Here is Harry Levin, writing about 35 years ago in his General Introduction to The Riverside Shakespeare:
The figure of Shakespeare as a practical man of affairs, although well attested by the evidence, seemed rather too modest to occupy the lofty pedastel erected by the Bardolaters. Hence the strange proliferation of irresponsible theories proposing rival candidates for the authorship of Shakespeare's work, most of them titled and all of them colorful but none of them circumstanced to have done the job--as William Shakespeare indubitably was.
Still true, and the proliferations continue apace.
The principle of Occam's razor--simple explanations are to be preferred to more complicated ones--recommends we conclude that Shakespeare wrote Shakespeare's plays. The alternative view requires too much intrigue, too many conspirators. The reason the editors of the First Folio attributed the plays to their friend and contemporary, William Shakespeare, is that he really did write them. Ben Jonson's famous poetic tribute was just that, a poetic tribute to the author, not an exercise in advancing a very intricate web of intrigue and deceit.