Here are few more details for David's answer.

In the case of rank 2 holomorphic vector bundles over a compact Riemann surface $X$, a maximally unstable bundle is a bundle $V$ where the slope (in this case, the degree) of a destabilizing line subbundle $L$ in $V$ is maximal possible. In the case when $V$ is flat this just means that $L$ is a square root of the canonical bundle $K$ of $X$. The details were worked out by Robert Gunning in "Special Coordinate Coverings of Riemann Surfaces" Math. Annalen 170, p. 67 - 86 (1967). Faltings in "Real projective structures on Riemann surfaces", Compositio Math. 48 (1983), no. 2, 223–269, looked at the case of punctured Riemann surfaces. I did not see any detailed analysis in the case when $X$ is replaced by a singular stable curve, but it is probably not too hard.

To relate (in the compact case) maximally unstable bundles to monodromy, suppose that $\rho: \pi=\pi_1(X)\to PSL(2, {\mathbb C})$ is a representation equivariant with respect to a (locally injective) holomorphic function $f: \tilde{X}\to CP^1$ defined on the universal cover of $X$. One should think of $f$ as a holomorphic section (again denoted $f$) of the $CP^1$-bundle $P\to X$ associated to $\rho$. Local injectivity of $f$ translated to the fact that $f$, as a section, is transversal to the leaves of the flat connection on $P$.

A representation $\rho$ as above is known to lift (nonuniquely) to a representation $\tilde\rho: \pi\to SL(2, {\mathbb C})$. Let $V\to X$ be the associated (to $\tilde\rho$) flat rank 2 holomorphic vector bundle. The section $f$ lifts to a holomorphic line subbundle $L\subset V$. Then one does a computation and shows that $L$ is destabilizing and half-canonical, i.e., $L^2\cong K$. This procedure can be reversed, namely, a maximally destabilizing line subbundle in a rank 2 vector bundle projects to a section of a $CP^1$ bundle over $X$, etc.

To relate to Schwarzian, note that the function $f:\tilde{X}\to CP^1$ has a well-defined Schwarzian derivative $S(f)$ which projects to a holomorphic quadratic differential $q$ on $X$. (Holomorphicity of $q$ is equivalent to local conformality of $f$.) Then $\rho$ is the monodromy of the Schwarzian differential equation $S(f)=q$ where we now think of $f$ as a multivalued function on $X$. Conversely, given a holomorphic quadratic differential $q$, one recovers $f$ (uniquely up to postcomposition with elements of $PSL(2, {\mathbb C})$), representation $\rho$ etc.