Context : Low-mass ( M _ { \star } / M _ { \sun } \lesssim 0.45 ) white dwarfs , including the so called extremely low-mass white dwarfs ( ELM , M _ { \star } / M _ { \sun } \lesssim 0.18 - 0.20 ) , are being currently discovered in the field of our Galaxy through dedicated photometric surveys . The fact that some of them pulsate opens the unparalleled chance for sounding their interiors . Aims : We present a detailed nonadiabatic pulsational analysis of such stars employing full evolutionary sequences of low-mass He-core white dwarf models derived from binary star evolution computations . The main aim of this study is to provide a detailed description of the pulsation stability properties of variable low-mass white dwarfs during the terminal cooling branch . Methods : Our nonadiabatic pulsation analysis is based on a new set of He-core white-dwarf models with masses ranging from 0.1554 to 0.4352 M _ { \sun } derived by computing the non-conservative evolution of a binary system consisting of an initially 1 M _ { \sun } ZAMS star and a 1.4 M _ { \sun } neutron star . We have computed nonadiabatic radial ( \ell = 0 ) and nonradial ( \ell = 1 , 2 ) g and p modes to assess the dependence of the pulsational stability properties of these objects with stellar parameters such as the stellar mass , the effective temperature , and the convective efficiency . Results : We found that a dense spectrum of unstable radial modes and nonradial g and p modes are driven by the \kappa - \gamma mechanism due to the partial ionization of H in the stellar envelope , in addition to low-order unstable g modes characterized by short pulsation periods which are significantly excited by H burning via the \varepsilon mechanism of mode driving . In all the cases , the characteristic times required for the modes to reach amplitudes large enough as to be observable ( the e -folding times ) are always shorter than cooling timescales . We explore the dependence of the ranges of unstable mode periods ( the longest and shortest excited periods ) with the effective temperature , the stellar mass , the convective efficiency , and the harmonic degree of the modes . We also compare our theoretical predictions with the excited modes observed in the seven known variable low-mass white dwarfs ( ELMVs ) , and found an excellent agreement . Conclusions :