  One of the important objectives of the COMPASS experiment (SPS, CERN)
\cite{Abbon:2007pq} is the exploration of the transverse spin structure of the
nucleon via spin dependent azimuthal asymmetries in single-hadron production in
deep inelastic scattering of polarized leptons off transversely polarized
targets. For this purpose a series of measurements were made in COMPASS, using
160 GeV/c longitudinally polarized muon beam and transversely polarized $^6LiD$
(in 2002, 2003 and 2004) and $NH_3$ (in 2007 and 2010) targets.
  In the past few years considerable theoretical interest and experimental
efforts were focused on the study of Collins and Sivers transverse spin
asymmetries. The experimental results obtained so far play an important role in
the general understanding of the three-dimensional nature of the nucleon in
terms of transverse momentum dependent parton distribution functions.
  In addition to these two measured leading-twist effects, the SIDIS
cross-section includes six more target transverse spin dependent azimuthal
asymmetries, which have their own well defined leading or higher-twist
interpretation in terms of QCD parton model.
  COMPASS preliminary results for these six "beyond Collins and Sivers"
asymmetries, obtained from transversely polarized deuteron and proton data have
been presented at the previous conferences \cite{Parsamyan:2007ju} -
\cite{Parsamyan:2013ug}. In this review we focus on the results obtained with
the last "proton-2010" data sample.
