Saint Ann Street is in the centre of Salisbury. It runs from the Cathedral close up towards Saint Martins. Saint Ann Place is just off of St Ann Street.
Which Saint Ann was the Salisbury street named after
There are two Saint Ann(e)s listed in the Catholic Encyclopedia:
Both St Ann Street and St Ann Place would be named after Saint Ann, the mother of Mary. St Ann Street leads to St Ann’s Gate which is one of the four gates to the Cathedral Close. The Close wall (and Saint Ann’s Gate) was built in the fourteenth century 1. Saint Anne Line was not born until the 16th century, and not canonized until 1970, so could not be the inspiration for St Ann Street.
The naming of St Ann’s Gate is appropriate in that the Cathedral is dedicated to her daughter Mary.

Reproduced courtesy of Francis Frith.
The possessive ‘S’ in Salisbury street names
Google Maps list Saint Ann Street as such, and not as Saint Ann’s Street. This does agree with what I think I’ve heard people in Salisbury saying.
Conversely, insofar as people refer to the Gate, I’m fairly sure I’ve only heard it called Saint Ann’s Gate, never St Ann Gate
I don’t know why most of the other roads named after saints have the possessive apostrophe-’s’ and Saint Ann Street doesn’t. With the exception of St John Street, the other roads which don’t have the ‘s’ either have a name which ends in ‘s’ e.g. Saint Thomas or sound like an ‘s’ e.g. St Lawrence. St John Street meets St Ann Street at St Anne’s Gate, so presumably it was named at roughly the same time – perhaps the apostophe-’s’ came into fashion at a later time?
[...] St Ann Street [...]
[...] Gate, another of the four gates in the Close wall. It’s at the junction of Exeter Street, St Ann Street and St John [...]