Elizabeth I's handwriting was typically a style known in the early modern period as "italic," in the literal sense that it originated in Italy [1]. Slight aside (print history is more my specialty than paleography): italic handwriting isn't directly related to the print typography style known as "italic," which was developed from earlier humanistic scripts by the Venetian printer Aldus Manutius circa 1500 [2]. If you look at Elizabeth's italic hand, it's remarkable how readable and clear it is, especially compared to other types of early modern English paleography. It would have been a signifier not only of her status, but also of her cosmopolitan humanist education.

Just like how in living memory there were different styles or "methods" for handwriting, like the Zaner-Bloser hand my grandmother used, in the early modern period there were different styles. In the English context, the two best-known are secretary hand and the slightly earlier (late medieval) chancery hand. Starting around the turn of the 16th century, i.e., a few generations before Elizabeth, aristocrats began using the italic hand more than secretary, but both and especially the latter were quite common among the literate, be they aristocrats or bourgeois.

Because of the popularity of genealogical research, there are actually a lot of decent paleographical resources on the Web [3, 4].

Sources:

[1] "Document 1: A letter dated 16 March 1554 from Elizabeth I, as princess, to her sister Queen Mary I," UK National Archives.

[2] Carter and Barker, ABC for Book Collectors

[3] Secretary hand tutorial

[4] List of online resources for early modern English paleography, Folger Shakespeare Library