  Certification of quantum systems and their properties has become a field of
intensive studies. Here, taking advantage of the one-sided device-independent
scenario (known also as quantum steering scenario), we propose a self-testing
scheme for all bipartite entangled states using a single family of steering
inequalities with the minimal number of two measurements per party. Building on
this scheme we then show how to certify all rank-one extremal measurements,
including non-projective $d^2$-outcome measurements, which in turn can be used
for certification of the maximal amount of randomness from every entangled
bipartite state of local dimension $d$, that is, $2\log_2d$ bits. Finally, in a
particular case of $d=3$, we extend our self-testing results to the fully
device-independent setting.
