Do spiders ever take up residence in abandoned webs?
The webs themselves? Not so much but that super awesome bug catching spot with what looks like an abandoned web already in it? Absolutely!
They are much more likely to simply build over an existing web or just take the awesome spot instead of reusing a web.
Spider's web are specific (like a fingerprint). They are fragile structures that need constant care and reparations, they won't last long (there are spiders that make stronger webs but still not enough resistant to several days of band conditions).
It's also important to point that some spiders are eaten by other spiders so it would be very risky to use an unknown web. Also, species of spiders use webs for different purposes (to live underwater too!), so they build them accordingly to their particular way of living (some to hunt, some to have a nice house).
I'll add, a lot of web weaving spiders eat and rebuild their webs daily to keep it clean. You can often find piles of dead bug sacs underneath orbweavers. Its really unsettling, but interesting. It would be odd for an orbweaver to abandon its web instead of eating it and moving on.