"Venus and Adonis" was dedicated, in the original, to "Henrie Wriothesley, Earle of Shouthampton" by "William Shakespeare" on April 18, 1593. On May 9, 1594, "The Rape of Lucerce" was dedicated to the same person by "William Shakespeare." (Dates were established "as entered for printing.")

On August 29, 1597, "The Tragedie of King Richard the second. As it hath beene publikely acted by the righ Honourable the Lorde Chamberlaine his Seruants" was entered for publication by Andrew Wise. No author was given to the first edition, but in 1598 the work was twice reprinted with the author listed as "William Shake-speare." On October 20, Wise entered "The Tragedy of King Richard the Third. Containing, His treacherous Plots...." Again, there was no author listed for the first edition, but in the second edition, appearing the following year, the author is listed as "By William Shake-speare."

On February 25, 1598, Wise entered for publication "The historye of Henry the iiij..."; no author given in the 1598 edition, the second edition of 1599 includes "Newly corrected by W. Shake-speare."

On September 7, 1598, Francis Meres entered for publication his "Palladis Tamia: Wit's Treasury." In it, he writes

...As Plautus and Seneca are acounted the best for Comedy and Tragedy among the Latines: so Shakespeare among the English is the most excellent in both kinds for the stage; for Comedy, witnes his Gentlemen of Verona, his Errors, his Loue Labors lost, his Loues labours wonne, his Midsummers night derame, & his Merchant of Venice: for Tragedy his Richard the 2. Richard the 3. Henry the 4. King Iohn, Titus Andronicus and his Romeo and Juliet.

At various times, "A Midsummer Night's Dream," "The Merchant of Venice," "The Merry Wives of Windsor," "The Tragical History of Hamlet, Prince of Denmark" and others were entered for publication with in inclusion of a phrase such as "by William Shake-speare".

I think that the contemporary evidence is quite good, actually, that William Shakespeare of Stratford-on-Avon wrote the plays he's generally credited with writing.