the surname is recorded from 1248; it means "a spearman." This was a common type of English surname: Shakelance (1275), Shakeshaft (1332), etc. To shake (v.) in the sense of "to brandish or flourish (a weapon)" is attested from late Old English:
Heo scæken on heore honden speren swiðe stronge.
[Laymon, "Brut," c. 1205]
and was in use through Middle English. Compare also shake-buckler "a swaggerer, a bully;" shake-rag "ragged fellow, tatterdemalion," an old name for a beggar.
"Never a name in English nomenclature so simple or so certain in origin. It is exactly what it looks -- Shakespear" [Bardsley, "Dictionary of English and Welsh Surnames," 1901]. Nevertheless, speculation flourishes.
The spelling is that of the first folio. The name was variously written in contemporary records, as all surnames were. In one signature, the author spells it Shakspere. It also was spelled Shakespear and Shakespere, the former being the proper modern spelling, the latter being the spelling adopted by the New Shakespere Society of London and the first edition of the OED and Century Dictionary. Related: Shakespearian (1753); Shakesperean (1796); Shakesperian (1755).