We further develop recently proposed new approach to description of the relativistic neutrino flavour νLe↔νLμ \nu_e^L\leftrightarrow\nu_{\mu}^L , spin νLe↔νRe \nu_e^L\leftrightarrow\nu_e^R and spin-flavour νLe↔νRμ \nu_e^L\leftrightarrow\nu_{\mu}^R oscillations in a constant magnetic field that is based on the use of the exact neutrino stationary states in the magnetic field. The neutrino flavour, spin and spin-flavour oscillations probabilities are calculated. In general, the obtained expressions for the neutrino oscillations probabilities exhibit new inherent features in the oscillation patters that are missing when the customary approach (based on the use of the neutrino helicity states) is used. It is shown, in particular, that in the presence of the transversal magnetic field for a given choice of parameters (the energy and magnetic moments of neutrinos and strength of the magnetic field) the amplitude of the flavour oscillations at the vacuum frequency is modulated by the magnetic field frequency.