I just experienced having a tire explosion yesterday. I was running at around 60mph and suddenly my rear right tire blew. I was go up a bridge and as it blew, there was brown smoke and I might have ran to around 50m before coming to a full stop. when I got out, I checked the tire and the rim, but the rim seems to be fine. I am concerned about what ever damages it might have done to the other parts(suspension,brakes, axles, etc). How do I check it personally before I bring it to my mechanic so at least I have an idea if there is something damaged?(I'm bringing it to my mechanic for a power steering repair, which is a different issue, and the mechanic does not specialize in suspensions and wheels.)
It's unlikely that there was any sort of shrapnel damage to the vehicle from a blown tire. It's more likely that the shredding tire whacked the vehicle a few times as it rotated to a stop.
I would expect to see cosmetic damage around the bumped and underbody: e.g., rubber streaks and dents. You might find bits of rubber jammed up underneath suspension bits. You'll also likely see damage to the rim if it was supporting the weight of the car.
In general, unless the tire completely delaminated and turned into a road gator, a blown tire is scary but doesn't damage the vehicle per se.