  The frustrated one-dimensional (1D) quantum magnet LiCuSbO$_4$ is one rare
realization of the $J_1-J_2$ spin chain model with an easily accessible
saturation field, formerly estimated to 12~T. Exotic multipolar nematic phases
were theoretically predicted in such compounds just below the saturation field,
but without unambiguous experimental observation so far. In this paper we
present extensive experimental research of the compound in the wide temperature
(30mK$-$300K) and field (0$-$13.3T) range by muon spin rotation ($\mu$SR),
$^7$Li nuclear magnetic resonance (NMR) and magnetic susceptibility (SQUID).
$\mu$SR experiments in zero magnetic field demonstrate the absence of long
range 3D ordering down to 30mK. Together with former heat capacity data [S.E.
Dutton \emph{et al}, Phys. Rev. Lett. 108, 187206 (2012)], magnetic
susceptibility measurements suggest short range correlated vector chiral phase
in the field range $0-4$T. In the intermediate field values (5$-$12T), the
system enters in a 3D ordered spin density wave phase with 0.75$\mu_B$ per
copper site at lowest temperatures (125mK), estimated by NMR. At still higher
field, the magnetization is found to be saturated above 13T where the spin
lattice $T_1^{-1}$ relaxation reveals a spin gap estimated to 3.2(2)K. We
narrow down the possibility of observing a multipolar nematic phase to the
range 12.5$-$13T.
