I will present some recent joint work with Sorin Popa where we show that, under the continuum hypotheses, any ultraproduct II$_1$ factor contains more than continuum many mutually disjoint singular MASAs. In other words, the singular abelian rank of any ultraproduct II$_1$ factor $M$, $\text{r}(M)$, is larger than $\mathfrak{c}$. Moreover, if the strong continuum hypothesis $2^\mathfrak{c}=\aleph_2$ is assumed, then $\text{r}(M) = 2^\mathfrak{c}$. More generally, these results hold true for any II$_1$ factor $M$ with unitary group of cardinality $\mathfrak{c}$ that satisfies the bicommutant condition $(A_0'\cap M)'\cap M=M$, for all $A_0\subset M$ separable abelian.