Let $f:[0,1]\to [0,1]^2$ be some continuous surjective function, and $\mu_n$ - the canonical Lebesgue measure on $\Bbb R^n$. So, on $[0,1]^2$ we can define two probability measures: $\mu_2$ and $\nu_2:=f_*\mu_1$ is the pushforward measure: $\nu_2(A) = \mu_1(f^{-1}(A))$ for each Borel-measurable set $A$.
My guess would be that they are mutually singular for all $f$, and that it shold not be too hard to find example sets on which each of them takes $0$ values while the other assigns positive measure, however I have not practiced measure theory for a while, so can't think of anything simple.
Question: I would like to know whether $\mu_2$ and $\nu_2$ are indeed mutually singular for any $f$ as above.
The answer depends on the choice of $f$. If $f: [0, 1] \to [0, 1]^2$ is the Hilbert or the Peano space filling curve, then it is measure preserving. You can also get singular measures by composing the Cantor-Lebesgue function with Hilbert/Peano curve. A good reference for these facts is "Space filling curves" by Hans Sagan.