I've owned a few of each and think that the 580 series was a significant leap forward accuracy wise from the 510 series.
When Remington made the 510 series, they were competing head to head against better designs from Winchester like the Models 69 and 72.
Remington introduced the 580 series around the time that Winchester dropped the Models 69 and 72, replacing them with the Models 121/131/141 .22 bolt guns. Even the most loyal Winchester man will probably agree that they were a giant step back from the quality of the 69 and 72.
The 580 series triggers are among the best any American manufacturer has put of a basic sporter style rimfire bolt gun..
260Remguy,
It isn’t very often that I “disagree” with you but I’ve never met a 580 series Remington that could hold a light to the few 513S sporters that I’ve had over the years (the 513s that I’ve had will shoot with some of the best .22 sporters I have including Cooper, Anschutz, and others of that ilk) - the 580 series might have a edge in the trigger department but that’s only because the 510 series triggers are notoriously bad. That said 580 series triggers can be altered to be really good triggers but stock they aren’t much to write home about - now 540 series triggers are a whole other ballgame. The 580 series rifles I have have/had all shot pretty good (around 1/2” at 50 yards) but can typically be made to shoot better with a tuned trigger.
That’s just my limited experience (sample size) obviously others may have had different results.
PennDog
