Bookshelf speakers for classical music

For $500 or less, I’d look at Wharfedale brand. They sound much better than their price might suggest. Many of the previous suggestions are above your budget, new.

New, Focal Chorus 706 bookshelf.  Not the most refined, but decent enough with front port.  Used,  small Spendors.  Up the ante and some small floor stander Spendors.  Used Rega monitor or small floor standers.

Vandersteen VLR's will play any music you throw at them.
And, do it well. But, they need to be placed near a corner to enhance bass response, though you can always add subs, like I do.
Bob

I remember an ad for the Rogers LS3/5A showing them placed on a bookshelf. This was back in '76, when they were only $450 a pair! Wish I had bought a pair or two then! Now look at what they cost! Maybe I was lucky to snag a pair for $960 ( from Lithuania, of all places!). 

Not much to go on, but here's a nice pair of Silverline Minuet Supreme Plus in your price range.  Great sounding monitors that don't require gobs of power to sound good.  Best of luck. 

https://www.ebay.com/itm/283583900981?ul_ref=https%253A%252F%252Frover.ebay.com%252Frover%252F1%252F...

https://www.ebay.com/itm/283583900981?ul_ref=https%253A%252F%252Frover.ebay.com%252Frover%252F1%252F711-53200-19255-0%252F1%253Ficep_ff3%253D2%2526pub%253D5575084939%2526toolid%253D10001%2526campid%253D5338381866%2526customid%253Dfd9d0170-cad3-11e9-8c3c-366639333233%2526icep_item%253D283583900981%2526ipn%253Dpsmain%2526icep_vectorid%253D229466%2526kwid%253D902099%2526mtid%253D824%2526kw%253Dlg%2526srcrot%253D711-53200-19255-0%2526rvr_id%253D2092424250874%2526rvr_ts%253De08284f716c0aadb22a5e56efff3945d&ul_noapp=true

I think classical really sounds good with wide, uniform soundstage. Maybe JBL or Kef.

Try fritz

I don't know any speakers that will stop working when you play classical music. 

You need to borrow a few pairs and try them at home. That is the best advice you'll get.