I have financed 2 cars with VW Credit in the past 4 months with a score of 670, and never had to show a paystub or any other proof of income.
I was instantly approved with BoA online a few days ago and the woman said all she needs to see is my purchase agreement...and this is even after I put in "1 month" with current employer and was approved for the lowest rate (2.09% for 72 months) at the full amount I asked for. My score is 774 according to their Equifax pull though.
I financed a brand new Acura at 0.9% for 60 months with American Honda Finance last August. At the Acura dealership, I was prepared to provide paystubs since I've never financed through Acura/Honda before. I pulled them out of my purse to offer them and the finance guy told me they weren't needed. He didn't even look at them. They pulled an EQ score of 777.
I think if your score is high or if you have financed through that lender previously with no negative marks, chances are that they won't want to see proof of income. But as always, YMMV.
I only make $23K / year and i didn't have to show any proof of income when i purchased my new car last month thru Hyundai with a 723 EX. 2 years prior when i tried to get a Corolla S they did want paystubs. I had high DTI and mid 600's score so maybe that was why.
I have bought many many cars. I have had to show a paystub one time. This was when I bought a car 4 weeks post BK 7 Discharge. All other deals I have not had to show any proof of anything.