I'd do a product like IronX on most of the Car. I had spots like that showing up on my plastic bumper a year after I bought my Sonic. Sprayed IronX on the car and hundreds of purple spots showed up (indicating where it is working). Let that sit for 5 minutes and rinse off. I'd bet your car brand...
Is the paint bubbled? This sort of looks like rail dust. Considering the spolier requires drilling to install, its possible that a metal flake got embedded in the paint and started to rust. A product like Iron-X would take care of that quickly.