Aims: A recent analysis of polarization measurements of HINODE SOT/SP in the quiet Sun pointed out very complex shapes of Stokes V profiles. Here we present the first classification of the SOT/SP circular polarization measurements with the aim of highlighting exhaustively the whole variety of Stokes V shapes emerging from the quiet Sun. Methods: k-means is used to classify HINODE SOT/SP Stokes V profiles observed in the quiet Sun network and internetwork (IN). We analyze a 302 × 162 arcsec2 field-of-view (FOV) that can be considered a complete sample of quiet Sun measurements performed at the disk center with 0.32 arcsec angular resolution and 10-3 polarimetric sensitivity. This classification allows us to divide the whole dataset into classes, with each class represented by a cluster profile, i.e., the average of the profiles in the class. Results: The set of 35 cluster profiles derived from the analysis completely characterizes the SOT/SP quiet Sun measurements. The separation between network and IN profile shapes is evident - classes in the network are not present in the IN, and vice versa. Asymmetric profiles are approximately 93% of the total number of profiles. Among these, about 34% of the profiles are strongly asymmetric, and they can be divided into three families: blue-lobe, red-lobe, and Q-like profiles. The blue-lobe profiles tend to be associated with upflows (granules), whereas the red-lobe and Q-like ones appear in downflows (intergranular lanes). Conclusions: These profiles need to be interpreted considering model atmospheres different from a uniformly magnetized Milne-Eddington (ME) atmosphere, i.e., characterized by gradients and/or discontinuities in the magnetic field and velocity along the line-of-sight (LOS). We propose the use of cluster profiles as a standard archive to test inversion codes, and to check the validity and/or completeness of synthetic profiles produced by MHD simulations.