in terms of imams, it depends on the branch of islam. some imams are just "knowlegdable" people in the mosque, whiilst in other branches they are a cleric and thus religious leader. the only 'qualification' you would need is to know (or claim to) more of the koran than anyone else in the congregation. oficially it is generally any respected muslim who is normally well-trained in leading prayer, salah, can be an imam. and he must have a beard.
priests (roman cath. etc) go through a process of holy orders to a point of ordination. a trainee priest must undertake five years of formation in seminary (more often a degree or suchlike in theology), and also spend time working in pastoral situations. the clergy who are mentoring the priest will make recommendaitons to the bishop who will then decide himself whether the candidate should be ordained as a deacon for 6 months, before being confrmed as a priest.
similarly, rabbis would aim for semikhah, which is also ordination. this requires theological study and indepth knowledge of the talmud and jewish law. mostly the semikhah is a formal qualification from a rabbinical institution or academy - such as the london school of jewish studies. the process takes 5 - 7 years.
under c of e there is a list of set criteria and following a similar vein to the catholics recommendaiton is made for a counicl to decide.
none of the above are approved by the government so essentially they are self-certifying. the main difference is c of e being a state church, where an arch bishop is appointed by the prime minister - or recommendation via the c of e appointents committee which the prime minister can reject - as thatcher did once. then recommendation is made to the sovereign for formalities to ensue in the college of canons.