Especially at Low Mass, sometimes the overeager faithful reply Amen when they ought not.
The priest begins the prayers at the foot of the altar by making the sign of the Cross, saying In nomine Patris, et Filii, et Spiritus Sancti. Amen. No one, neither the server(s) nor the rest of the congregation, replies here - the priest himself says the Amen.
When the priest recites the Introit, he reads the Gloria Patri, ending with an Amen; however, as he immediately repeats the Introit antiphon, it is far less likely that an overzealous attendee would attempt to say Amen at that point.
It should be carefully noted, though, that when the priest recites the Gloria in excelsis, and, likewise, the Credo, in both cases he himself concludes each with an Amen, which is his to say, and is not the province of the faithful present at Mass, however heartily they concur – despite this, some mistakenly try to reply Amen, not realising that here there is no reply to give.
When the priest administers Communion to the faithful, he says to each Corpus Domini nostri Jesu Christi custodiat animam tuam in vitam æternam. Amen. Again, the individual communicant ought say nothing – as they are at that moment kneeling with their tongue extended, ready to receive the Host, by trying to saying Amen, they will only obstruct the administration of Communion.
It is always wise to be aware of the rubrics.