With characteristic charm our beloved Holy Father has described priests who refuse to baptise the children of single mothers as 'animals'. I would not refuse baptism for that reason, I do say that I expect to see parents or parent at Mass, that is about the only criteria I ask for.
My friend Fr Alexander Lucie Smith makes the very reasonable point:
By the time the engaged couple present themselves, saying they want to get married in Church, it is already too late to start preparation. Preparation for marriage needs to start before the couple have met; it needs to start in childhood, or even, to be on the safe side, at birth. After all, whom you marry is the most important decision you will ever make.The problem is that the child of a parent who doesn't practice or live according to the Church's teaching -sometimes actually rejecting large portions of it- is unlikely to practice the faith or live it. Statistics prove that they are themselves unlikely to be able to make a stable marriage themselves, thus in a sense proving the Pope's second point which simply rocks the whole structure of Catholicism, that majority of marriages are invalid.
Perhaps we really do need to be more rigorous in our baptismal preparation. Though we might not decline to baptise, perhaps we should defer it. Perhaps using the baptism of children to encourage parents to marry, to form a stable relationship in which to educate or bring up a child is not such a bad thing.
Maybe the Church, and couples but especially children might be better served by a few animals.