Usually a SP. I think what they mean by "obtaining a consumer credit report" is that they will pull you but on the secondary bureaus such as Chexsystems or something, not the usual EX/EQ/TU credit bureaus.
It's good to know that most banks only do a SP now... when I was opening new accounts years ago that wasn't the case, at least not locally.
These are 'stale' data points, but my experience back in 2000 with BBVA/Compass was a HP for a checking account; a friend who was an employee told me that if you qualified for a credit card they would offer you one at that time as well (it wasn't mentioned to me, so apparently I didn't qualify).
I opened a checking account with Wachovia (now Wells Fargo) in 2005; it was a HP as well. I don't know if they ran a pre-qual at the time, but they did note that my score was 'good', and volunteered to increase my ATM daily limit because of it (I later found out they would do this for nearly anybody who asked, but at the time it was an ego-boost).
My most recent checking account opening was/is with USAA, opened in early 2012. They didn't pull credit from any of the major bureaus, and I'm not aware of any secondary bureau they might have used. It's said that they don't use ChexSystems (it wasn't a concern of mine, but it came up repeatedly when I was researching them back then, since they were open to the public at the time).
I closed my BBVA/Compass account back in 2013. I hadn't actively used it in years; the only reason I kept it was because it was free, and they reimbursed the ATM fees charged by other bank's machines - but those both changed. They added a monthly fee then added things could do to keep it 'free', and they did away with their ATM fee reimbursement feature, which eliminated my need for them. After they did away with 'free' accounts, for a couple of years I kept $1500 on deposit just to waive the fee, but eventually I needed the cash, so I had them close the account. Ironically, twice since then (the last time was earlier this week) I've received an email from them asking me to 'come back', that they are once again offering a 'free' account. It's nice to feel wanted, but I think my needs are being met quite well where I am... LOL
Those three are the only checking accounts I've opened since 2000. I had three or four during the '90s when I was still figuring things out, and I co-signed for a couple of my younger brothers a time or two (also in the '90s) but my experience with checking accounts has slowed down considerably since then. My income is rather modest, so it would be a stretch to keep more than two active at a time anyway. At one time my checking account statements were multi-page, but now both are just a single page (sometimes two if there's a generic 'notice' attached).
All that being said, if I were in the market for a new checking account, I would certainly want to get as much 'bang for the buck' as possible out of the deal, be it SP or HP! 