  The low energy excited $0_{2,3}^+$ states in $^{96}$Sr are amongst the most
prominent examples of shape coexistence across the nuclear landscape. In this
work, the neutron $[2s_{1/2}]^2$ content of the $0_{1,2,3}^+$ states in
$^{96}$Sr was determined by means of the d($^{95}$Sr,p) transfer reaction at
the TRIUMF-ISAC2 facility using the SHARC and TIGRESS arrays. Spectroscopic
factors of 0.19(3) and 0.22(3) were extracted for the $^{96}$Sr ground and
1229~keV $0^+$ states, respectively, by fitting the experimental angular
distributions to DWBA reaction model calculations. A detailed analysis of the
$\gamma$-decay of the isomeric $0_3^+$ state was used to determine a
spectroscopic factor of 0.33(13). The experimental results are compared to
shell model calculations, which predict negligible spectroscopic strength for
the excited $0^+$ states in $^{96}$Sr. The strengths of the excited $0_{2,3}^+$
states were also analyzed within a two-level mixing model and are consistent
with a mixing strength of $a^2$=0.40(14) and a difference in intrinsic
deformations of $|\Delta \beta|=0.31(3)$. These results suggest coexistence of
three different configurations in $^{96}$Sr and strong shape mixing of the two
excited $0^+$ states.
