Only one artist has actually recorded all of these sonatas, the very gifted harpsichordist Scott Ross, who passed away tragically young. This is him playing the Sonata K. 27 in B minor:
Here is guitarist John Williams playing my favorite sonata, K. 213 in D minor:
Composer/guitarist Leo Brouwer has recorded some of the most creative performances of Scarlatti sonatas. This is K. 206 in E major:
Going back to the keyboard, here is a very zippy performance of K. 141 in D minor by Martha Argerich:
And finally, returning to the harpsichord, this is Gustav Leonhardt with K. 227:
As Steven mentions in his blog post, Scarlatti, while greatly admired by both composers and performers, does not fit easily into the standard narrative. His musical imagination and style lies rather out of the mainstream, partaking of both Baroque and Classical elements without being firmly in either camp. As one writer noted in passing, Scarlatti was perfectly capable of inventing Classical style in a sonata or two, and then putting it aside to do something else! As W. Dean Sutcliffe mentions in his book (linked to by Steven), the critical landscape regarding Scarlatti is a vacant one. To put it bluntly, almost no work has been done in the critical appraisal and examination of Scarlatti's music.
But we can still enjoy it!