The only model-independent experiments dedicated to neutrino mass determination are the kinematic ones from single β-decay. In this context an international collaboration is growing around the project of Microcalorimeter Arrays for a Rhenium Experiment (MARE) for a direct calorimetric measurement of the neutrino mass with sub-electronvolt sensitivity. MARE is divided into two phases. The first phase consists of two independent experiments using the presently available detector technology to reach a sensitivity of the order of 1 eV, and to improve the understanding of the systematic uncertainties specific of the microcalorimetric technique. The two experiments are: MARE-1 in Milanin collaboration with NASA/GSFGthe University of Wisconsin at Madison, and MARE-1 in Genoa. The goal of the second phase (MARE-2) is to achieve a sub-electronvolt sensitivity on the neutrino mass. The Milan MARE-1 arrays are based on semiconductor thermistors and dielectric silver perrhenate absorbers, AgReO₄. To optimize the detector performance, crystals of silver perrhenate have been glued to the thermistors with different epoxy resins in order to determine the best thermal coupling. Now a 72 channel measurement is starting. To identify a shielding configuration that minimizes radioactivity background in the energy region of the ¹⁸⁷Re β spectrum, a preliminary study of the cryogenic laboratory environmental background has been performed. Using a planar germanium detector for the energy range below 10keV, different shielding configurations have been realized to find the best thickness and material to shield the microcalorimeter arrays.