Being from Ciren I don't have many unique Swindon memories, so my favourites from when I could come over on my own on the bus are:
The adver man....adddddva....addddddva

Beatties toy shop.
Rhapsody in the Brunel centre. Had a good tattoo shop in the basement.
Was it Rhapsody that also sol second hand records downstairs?
the fakey sport shop at the top of the brunel market was a highlight
The only thing I can think of adding is the dive of a video games arcade just round the corner from Beaties IIRC.
Good for filling in time for the bus home though.
And moving on in age, not quite as in famous as Goldiggers in Chippenham though!:
Cairos!
Route 66
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On a personal memory level, my and my mate still use it in the context of Swindon ladies now ( all in jest

)
Was walking up towards Lloyds bank, some schoolgirl got pushed over by another girl of a
similar age, who kicked her and ran off to the screams of "I'm going to knife you, you slag"..
Me and my mate looked at each other and said "yup, we're in Swindon"..