We present high-resolution triple-axis neutron scattering studies of the high-temperature superconductorLa1.88Sr0.12CuO4 (Tc = 27 K). The temperature dependence of the low-energy incommensurate magnetic fluctuations reveals distinctly glassy features. The glassiness is confirmed by the difference between the ordering temperature TN ≃ Tc inferred from elastic neutron scattering and the freezing temperature Tf ≃ 11 K obtained from muon spin rotation studies. The magnetic field independence of the observed excitation spectrum as well as the observation of a partial suppression of magnetic spectral weight below 0.75 meV for temperatures smaller than Tf, indicate that the stripe frozen state is capable of supporting a spin anisotropy gap, of a magnitude similar to that observed in the spin and charge stripe-ordered ground state of La1.88Sr0.12CuO4. The difference between TN and Tf implies that the significant enhancement in a magnetic field of nominally elastic incommensurate scattering is caused by strictly inelastic scattering-at least in the temperature range between Tf and Tc - which is not resolved in the present experiment. Combining the results obtained from our study of La1.88Sr0.12CuO4 with a critical reappraisal of published neutron scattering work on samples with chemical composition close to p = 0.12, where local probes indicate a sharp maximum in Tf(p), we arrive at the view that the low-energy fluctuations are strongly dependent on composition in this regime, with anisotropy gaps dominating only sufficiently close to p = 0.12 and superconducting spin gaps dominating elsewhere.