Most "may issue" statutes spell out a few (usually obvious) disqualifications, but the central issue that makes it a "may issue" state is that discretion is given to the issuing authority, which can be a judge, sheriff, or chief of police to reject the application. Most of the language has an escape clause that enables that authority to do so at will, typically that the applicant must show "need" or "necessity".
In most "may issue" states, it's highly variable, depending on the sheriff/judge whatever from one county to the next. In some, however, it's pretty universal. NJ, for example, handles these things at the county court level, and county judges, which are political appointees are unanimous in the notion that no normal citizen "needs" a carry permit, and that publicly armed citizens are "not in the public interest".