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..