My old lady dog hates being left alone in any sort of "confined" state (crate, car, room with door closed, etc), so when I leave her in the car, she usually starts whining within a few minutes, and screaming/barking within about 5-10 minutes. This prompts concerned people to stop and view her, and I know at least some of them must think she needs help (she may, but not of the sort that they're thinking). Additionally, because she freaks out, I have to worry that she would cause herself to overheat on even moderately warm days. She's a little better if the other dog is in the car, but unpredictable... she may be quiet for 20 minutes, or she may be screeching, panting, and acting a fool within 2 minutes, so she really can't be left unattended in a public setting for long. I once had her secured to the middle seatbelt of my Dad's truck with both windows down on an 80 degree evening (sun already set) while I ran in to grab a slice of pizza that the rest of my family was eating inside, and there were several people standing there considering action when I came out not 2 minutes later. Apparently they had heard her whining and even though she was essentially "under a tent", they thought she was in distress due to the heat. I thanked them for their concern, and assured them that she was just upset that I had left her alone.
My younger dog is awesome in the car, which is good because he is a complete brat when I leave him with my mom and go somewhere (alternative is to leave him in my bedroom at my house, which I feel bad about when he could be somewhere there are people and/or his dog buddy), so I take him with me a lot. I work a lot of evenings, so much of my errand running is at night, and he just curls up on the seat and sleeps til I come back. When I put the key in the door, he pops up, and I can't tell you how many times people parked nearby have said, "I never even saw a dog in there, and I've been here for XX minutes!". On sunny days he likes to get up on the dash and sunbathe, which I have tried to discourage to no avail.
If it's warm, I crate him in the back of the van (dark tinted windows) and leave the windows all open, park in shade, even if it's cloudy. My general rule it that if it's cool enough for me in there, it's ok for him, for short periods. I don't go in anywhere for long, and if it's going to be longer than I planned, I'll leave- I think that's where a lot of people get into trouble, they think, "It'll only be 5 more minutes"... meanwhile their dog is cooking. In cooler weather, I'll leave him for longer times, but if it's day time, or I'll be a while, I crate him so people can't see him. He's quiet as a mouse in the car, so I doubt people have any clue there's a dog in there.
I would say that on average, he's not typically in the car alone for more than 15-20 minutes at a time, but there are exceptions. Last year my cousin and I took him to a dog event in January (cold!), then on the way home stopped at my nephew's school sports match. I figured we'd be only an hour or so, but we were there a few hours. I had crated him, and he already had a big blanket in there, plus I covered his crate with another blanket. When I came out to check on him an hour into it (and a couple other times), he was sleeping, warm as toast, and perfectly content. We gave my nephew and one of his friends a ride home, and he scared them when he started wagging his tail after they sat down, they had no idea he was in there!
There's a laundromat near where I work, and a supermarket. I frequently see dogs waiting in cars alone at both, and have never seen one that looked physically distressed. I've seen dogs with heat "injuries" from being in hot cars, but IME there's definitely a middle ground where a dog can benefit and be perfectly safe riding along- like anything, it boils down to common sense and knowing your dog.