Continuity mistake: When Tommy Lee Jones pops into his local Irish boozer for a pint of what is supposed to be Guinness, it is clearly not a pint, it is clearly not Guinness (flat Coke, more like), and the level of said beverage in the glass fluctuates wildly and out of synch with TLJ's swigging.
Plot hole: The premise of the entire movie's main character, played by Jeff Bridges, is flawed, as it would be impossible for anyone to pass a major metropolitan police department's pre-employment screening background investigation for the position of police officer if that applicant uses a fake name and identity. Part of the investigation includes sending off the applicant's finger prints to the National Criminal Information Center, a polygraph test, a records check with vital statistics to verify place of birth, Dept. of Labor to verify past employment, Social Security Dept. and also interviewing (sometimes in person) your friends you've listed and your current and former neighbors of where you now live and use to live.