I realize the title is perhaps not the most helpful. I am aware of several uses of the word "trivial," and I'm hoping that perhaps someone can provide some further insight.
1) Trivial sub-objects, i.e. if $G$ is a group, then $1, G \leq G$ are trivial subgroups of $G$.
2) Trivial solutions to a problem, i.e. $y=y'$ has a trivial solutions as a differential equation, $y=0$.
3) Trivial factors of a number or polynomial, i.e. $1,n|n$ $ \forall n \in N$
What these two meanings have in common is that they are defined in this way. Trivial subgroups or solutions are well-defined ideas. But it seems to me that many times, the word is also used in association with proofs when referring to arguments that are simple or otherwise straight forward. It seems obvious that we would say that a proof of a simple theorem like "even's squared are even" or some such would be "trivial," but a quick look at the top unanswered questions right now reveals many problems that are not trivial at all. Why is this so? Is triviality just defined by common consensus that a problem is hard? If the issue is really so subjective, is the claim "all mathematics is trivial" just as valid as any other claim about it (even if this seems a bit ridiculous)?
I feel inclined to point out that I don't believe that mathematics is a trivial subject, I'm just trying to understand what this common expression really means. It seems sometimes it is quite easy for the reader, and sometimes quite difficult!