I understand most parts of the hiring process for an academic job, except for the reference check. How do they check it, who checks it, and when do they check it? Do they check what we write in the cover letter, cv, teaching/research statement, etc.? Do they run a background check, credit score check on you?
At my institution in the United States, reference checks are done by members of the search committee for the position, who call those people the candidate has listed as references. We use a structured list of questions, ending with, "is there anything we should know that I haven't asked you?"
We verify academic credentials by requiring official transcripts, including those from undergraduate institutions.
The institution separately does a criminal record background check, handled by the university police department. The candidate must explicitly authorize this, in writing. But, of course, we cannot consider candidates who will not agree to the background check. As far as I know, there is no credit check for faculty jobs. There probably is a credit check for anyone who handles money.
Finally, the institution does a driving record check every year. This is because we are covered by the university's insurance when driving university vehicles or driving our personal vehicles on university business.
What I've written (except the bit about the driving record) applies to people being hired into a tenure-track position. Applications for promotion or tenure for those already on the faculty are much different.
I don't think I'm the least-aware of bureaucratic activities at my university, but, for faculty or even postdoc positions, I have not been aware of any transcript-checking (either grad or undergrad) or phone-calling references or criminal background or driving record or credit checks.
It is conceivable that some of this is happening, but I've never heard a peep about it, while I know that some people who were hired had some credit problems, driving issues, and so on, so I don't know what, if anything, the central administration is doing with whatever info they do check.
I think essentially all the people we've hired for the 30+ years I've been here have had PhD advisors well-known to at least one of my colleagues... which, while not guaranteeing credit or driving virtues, guarantees something about mathematical competence.
I'm the chair of my (US public computer science) department's recruiting committee. At my university there are a few types of checks:
Recommendation letters. For assistant professor candidates, we request letters only from names that the applicant suggests in their application. But for anything more senior, we also request letters from people the applicant does not propose; in fact, a strict majority of the reference letters must come from people not suggested by the applicant.
Public academic record. We definitely look at the applicant's actual publications, but we may look at externally curated records of your publication, citation, and/or funding history. For example, in computer science, essentially all publications are indexed in DBLP, and citations are most accurately tracked by Google Scholar; we may check those sites. (Other fields use different databases.) Similarly, for tenured applicants, we may directly consult NSF's database of funded grant proposals. If an applicant has taught at an institution that publishes teaching evaluations, we may also look at those. We don't do these checks to verify factual information in the cover letter, CV, statements, and so on, but rather to get an independent and more consistent view of the applicants' records that helps us make direct comparisons.
Criminal background check. These are conducted by the state police on behalf of the university after an applicant accepts a contingent offer.
We do not request transcripts (even for assistant professors) or credit checks. Also, unlike Bob Brown's department/university, we do not call references for tenure-track faculty positions; we require all recommendations in writing.
Before we hire anyone for a tenured position, we not only consider the letters provided by the candidate's chosen references but we also request letters from additional experts in the candidate's research area. These experts are chosen by our department's personnel committee without consulting the candidate. No transcripts are involved; by the time someone is qualified for a tenured position, transcripts are ancient history. As for information in the candidate's CV, we generally presume that it's accurate unless we encounter evidence to the contrary. I believe the university conducts a background check, but I don't know how much it includes; my guess would be that they only look for any criminal records, not credit reports or driving records.