Are these group presentations trivial? I just got these presentations of groups:
$\langle a,b\mid aba^{-1}b^{-1}\rangle$
$\langle a,b\mid aba^{-1}b^{-2},bab^{-1}a^{-2}\rangle$
$\langle a,b\mid abab^{-1}\rangle $
Are any of them trivial? How do you prove it?
 A: Standard techniques that work with the kinds of groups you've given is to compute the abelianization of the group.  At least one of your groups is abelian - so you can describe it using the classification of finitely-generated abelian groups. 
Sometimes the abelianization gives you further ideas, like trying to prove your group is a semi-direct product.  There's something called the Reidemeister-Schreir technique that will help you with this.  Your 3rd group appears to be a semi-direct product.  
A: For the first and last group, try finding a homomorphism from your group $G$ to an abelian group.  If you can find an onto map $G \to A$ where $A$ is nontrivial, then $G$ is non-trivial.  If you have a group $G=\langle a,b\mid r\rangle$ and you decide you want to map $G\to A$ by $a\mapsto x$ and $b \mapsto y$, you get a homomorphism if and only if the result of substituting $x$ for $a$ and $y$ for $b$ in $r$ is trivial.
For the middle one, you've got $aba^{-1}=b^2$ and $bab^{-1}=a^2$ (1).  Rewriting the first one gives $ab (ba^{-1}b^{-1}) = b$.  Work out $ba^{-1}b^{-1}$ from (1) and substitute...
