Short answers, pulled from the story.
There are six surviving signatures written by Shakespeare himself, all on legal documents. They are spelled "Willm Shakp", "William Shaksper", "Wm Shakspe", "William Shakspere", "Willm Shakspere", and "By me William Shakspeare". None of the six uses the now-standard spelling "Shakespeare".
Shakespeare's six signatures appear on four documents: a deposition in the Bellott v Mountjoy case dated the 11th of May 1612; the purchase of a house in Blackfriars, London, dated the 10th of March 1613; the mortgage of that same house dated the 11th of March 1613; and his Last Will and Testament dated the 25th of March 1616, which carries three signatures, one on each page.
The spelling gained dominance largely through the authority of the Cambridge Shakespeare, the nine-volume edition edited by William George Clark, John Glover, and William Aldis Wright, published between 1863 and 1866. A companion Globe Edition issued in 1864 became so widely used that it remained a standard text for nearly a century, and the Cambridge spelling became the accepted form.
The antiquarian John Pinkerton revived the spelling "Shakspere" in 1785 in Letters on Literature, arguing it was the form traced by the poet's own hand in his signatures. The London publisher Bell gave it official status in 1788, and by the Victorian era it was used by many prominent scholars and institutions including the New Shakspere Society, founded in 1873 by Frederick James Furnivall.
Advocates of the Shakespeare authorship question used different spellings to distinguish between the supposed real author and the man from Stratford-upon-Avon. Some used "Shakespeare" or "Shake-speare" for the claimed hidden author and "Shaksper" or "Shakspere" for the Stratford man. H.N. Gibson noted that outlandish spellings were sometimes chosen purely to make the Stratford man appear vulgar or rustic.
Name spelling variation was common in the Elizabethan era. R.C. Churchill noted that Sir Walter Raleigh's contemporaries spelled his name in at least ten different ways, and Thomas Dekker's name appeared in at least eight variants. Kathman also observed that spelling tended to be more uniform in printed texts than in manuscripts, and more varied in provincial documents than in metropolitan ones.