We present the entire sample of ultraviolet (UV) spectra of supernovae (SNe) obtained with the Ultraviolet/Optical Telescope (UVOT) on board the Swift satellite during the first two years of observations (2005/2006). A total of 29 UV-grism and 22 V-grism spectra of nine SNe have been collected, of which six are thermonuclear (Type Ia) and three core-collapse (Type Ibc/II) SNe. All the spectra have been obtained during the photospheric phase. After a comparison of the spectra of our sample with those in the literature (SNe 1992A, 1990N, and 1999em), we confirm some degree of diversity in the UV emission of Type Ia SNe and a greater homogeneity in the Type II Plateau SN sample. Signatures of interaction between the ejecta and the circumstellar environment have been found in the UV spectrum of SN 2006jc, the only SN Type Ib/c for which UVOT grism data are available. Currently, Swift UVOT is the best suited instrument for early SN studies in the UV due to its fast response and flexible scheduling capabilities. However, in order to increase the quality of the data and significantly improve our understanding of the UV properties of SNe and to fully maximize the scientific potential of UVOT grism observations, a larger investment in observing time and longer exposures are needed.