We present results on neutron scattering in solid 4 He in the range of parameters where supersolidity is observed. The measurements address, among other questions, the viability of one possible mechanism of supersolidity: via a metastable amorphous phase. We have attempted to observe a glassy phase by neutron scattering. We have found that it is impossible to do this by total scattering, as it would be common in a classical solid, due to an extremely large inelastic diffuse signal related to the anomalously strong zero-point motion of helium atoms. This raises a general question on the interpretation of such scattering as the signature of an amorphous phase. Results from energy-resolved elastic scattering are heavily influenced by multiple scattering of neutrons which may be the major contribution to the measured elastic signal, but allow to put the limit on the concentration of an amorphous phase to 5% in a polycrystal with millimeter-size crystallites and to 2% in a single crystal. The values of NCRIf, expected from these limits should be much lower, although exact values depend strongly on a particular model of glass-related supersolidity.