If
X
X
is a separable Banach space, we consider the existence of non-trivial twisted sums
0
→
C
(
K
)
→
Y
→
X
→
0
0\to C(K)\to Y\to X\to 0
, where
K
=
[
0
,
1
]
K=[0,1]
or
ω
ω
.
\omega ^{\omega }.
For the case
K
=
[
0
,
1
]
K=[0,1]
we show that there exists a twisted sum whose quotient map is strictly singular if and only if
X
X
contains no copy of
ℓ
1
\ell _1
. If
K
=
ω
ω
K=\omega ^{\omega }
we prove an analogue of a theorem of Johnson and Zippin (for
K
=
[
0
,
1
]
K=[0,1]
) by showing that all such twisted sums are trivial if
X
X
is the dual of a space with summable Szlenk index (e.g.,
X
X
could be Tsirelson’s space); a converse is established under the assumption that
X
X
has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with
C
(
ω
ω
)
C(\omega ^{\omega })
with strictly singular quotient map.