Motivated by the recent discovery of rare Enormous Lyman-Alpha Nebulae (ELAN) around z ∼ 2 quasars, we have initiated a long-term observational campaign with the MUSE/VLT instrument to directly uncover the astrophysics of the gas around quasars. We present here the first 61 targets of our effort under the acronym QSO MUSEUM (Quasar Snapshot Observations with MUse: Search for Extended Ultraviolet eMission). These quasars are characterized by a median redshfit of z = 3.17 (3.03 < z < 3.46), absolute i magnitude in the range −29.67 ≤ M i (z = 2) ≤ −27.03, and different levels of radio-loudness. This sample unveils diverse specimens of Lyα nebulosities extending for tens of kiloparsecs around these quasars (on average out to a maximum projected distance of 80 kpc) above a surface brightness SB> 8.8×10 −19 erg s −1 cm −2 arcsec −2 (2σ ). Irrespective of the radio-loudness of the targets, the bulk of the extended Lyα emission is within R < 50 kpc, and is characterized by relatively quiescent kinematics, with average velocity dispersions of σ Lyα < 400 km s −1 . Therefore, the motions within all these Lyα nebulosities have amplitudes consistent with gravitational motions expected in dark matter halos hosting quasars at these redshifts, possibly reflecting the complexity in propagating a fast wind on large scales. Our current data suggest a combination of photoionization and resonant scattering as powering mechanisms of the Lyα emission. We discover the first z ∼ 3 ELAN, which confirms a very low probability (∼ 1%) of occurrence of such extreme systems at these cosmic epochs. Finally, we discuss the redshift evolution currently seen in extended Lyα emission around radio-quiet quasars from z ∼ 3 to z ∼ 2, concluding that it is possibly linked to a decrease of cool gas mass within the quasars' CGM from z ∼ 3 to z ∼ 2, and thus to the balance of cool vs hot media. Overall, QSO MUSEUM opens the path to statistical and homogeneous surveys targeting the gas phases in quasars' halos along cosmic times.