Twisted sums with C(K) spaces

If X is a separable Banach space, we consider the existence of non-trivial twisted sums 0\to C(K)\to Y\to X\to 0, where K=[0,1] or \omega^{\omega}. For the case K=[0,1] we show that there exists a twisted sum whose quotient map is strictly singular if and only if X contains no copy of \ell_1. If K=\omega^{\omega} we prove an analogue of a theorem of Johnson and Zippin (for K=[0,1]) by showing that all such twisted sums are trivial if X is the dual of a space with summable Szlenk index (e.g., X could be Tsirelson's space); a converse is established under the assumption that X has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with C(\omega^{\omega}) with strictly singular quotient map.