First, any description which doesn't uniquely specify the elements of a set tends to give a proper class.
For example, a group is an set together with operations satisfying a certain collection of axioms. Since these axioms don't pin down the elements of the group, one might expect the collection of all groups to not be a set.
This holds in a similar fasion for topological spaces, rings, vector spaces, fields, manifolds, metric spaces, and many of the other commonly studied objects.
This also applies to your "collection of all singletons" example, and similarly to any sort of "collection of all sets of a fixed size".
But just because a description DOES pin down the elements doesn't mean you have a set. For example, the collection of all ordinal numbers is not a set, despite the fact that if there is an order preserving bijection between two ordinals $\alpha$ and $\beta$, then one must have $\alpha = \beta$ as sets.
Similarly, the collection of all cardinals doesn't form a set.