Ok I should know this but it's got me confused.

[sparkybabe]

On my drive to work I drive up St Mary's Close and into Anthonys Close. One has an apostrophe and the other hasn't.

Why?

Are the streetsigns from different time periods? Is St Mary's from older, more pedantic times and Anthonys a modern aberration?

Are these signs in Birmingham perchance?

Perhaps the Anthonys were a family, and the Close is named after them, rather than a specific Anthony? That would mean it ought to be Anthonys' Close, but I can see why they might just skip the apostrophe there.

My street sign says St Wilfrids Road. It annoys me every day.

No not in Birmingham! It annoys me because one road leads into another, so they should either both have apostrophes, or neither!