$z \mapsto z^2$ is conjugated to $x \mapsto 2x$ mod 1 on ${\bf R}/{\bf Z}$. This map is in turn semiconjugated to the shift map $\sigma(\{a_n\}_{n\in {\bf N}} = \{a_{n+1}\}_{n\in {\bf N}}$ on $\{0,1\}^{\bf N}$ through base 2 decomposition.
$$
\{a_n\} \mapsto \sum_n {a_n \over 2^{n+1}}
$$
So you can push any ergodic probability measure invariant by the shift to get a continuous measure which is singular from Lebesgue as soon as it is not Lebesgue. So for example, consider for some $p \in (0,1)$ not equal to $1/2$,
$$
\mu_p = (p\delta_{0} + (1-p)\delta_{1})^{\otimes {\bf N}}
$$
and push this measure to the circle with the map
$$\varphi : \{a_n\} \mapsto exp\Bigl(2\pi i \sum_n {a_n \over 2^{n+1}}\Bigr)$$
There are many ergodic invariant probability measures by the shift which have full support. These are built using thermodynamic formalism. A reference is the book of Bowen, LNM 470, *Equilibrium states for Axiom A diffeomorphisms*. Any book about hyperbolic or symbolic dynamics should fit. Brin Stuck, *Introduction to dynamical systems*, Katok-Hasselblatt *Modern theory of dynamical systems* etc.