... and you'll know.

Seriously, that's a "sell by" date. The stuff lasts way longer than that.

It will turn green and furry. In time.

Seriously though, the pH of sour cream is not low enough to prevent bacterial growth or mold formation.
The proteins can be bacterially decomposed into a mines and formamides.
Fortunately, those smell bad so it's easy to know if it's bad. Plus as the milk proteins break down & the fat hydrolyzes it separates into curds & whey. The whey would show as a yellow, watery liquid on top.
And, mold is visible, even at very low levels because it's dark against the white product.
Yeah, it's already soured but no so far as to prevent microbes from thriving on the lactose in there.
It lasts a long time and is good at least 3 weeks after the sell-by date, but if definitely can spoil.