My HP Pavilion Slimline s7510n PC has no sound. I upgraded my PC and from vista to 10, lost my sound. The PC says I have a working digital audio (S/PDIF) (High Definition Audio Device) but no sound.

HP did not say you can do that go to w10. so what makes you thing a sound chip works.
no sound from internal speaker or head phones and earbuds?
the service tag shows the full model and p/n
the part number tells the helper what sound chip of many was used.
Windows 10, 32 bit or 64.?
2006 PC, 11 years old, most dont do 64 that old. ever.
nor on this OLD DOG Celeron M 380 (D) 1.6 GHz
HP relic pages, Legacy pages (terse) shows SOUND AS:
Realtek ALC 880 chipset

so lets to go realtek , HP does not support w10 on old old PCs.
sorry.
lets see of the chip maker supports w10 on this 11+ years old chip

here they are USA side,(not Taiwan)
http://www.realtek.com/contact/contentView.aspx?Langid=1&PNid=3&PFid=73&Level=2

shows nothing.
my guess no w10 drivers ever existed, nor will they
leaving (my guess 1, is Vista was last one)
trying any VISTA/W7/w8 drivers found, the MIGHT work.
or buying a USB sound dongle device, plug it in and have sound.

that is always to do that.
unless you can fit a new sound card in that slime line case.
id have to see inside it , go, will it fit or not. (size matters)