We present the results of Raman studies in the temperature range of 293-77 K on vibrational properties of linoleic and oleic acids and Raman microspectroscopy of human breast tissues at room temperature. Our results confirmed the significant role of unsaturated fatty acids in differentiation of noncancerous and cancerous breast tissues and the role of vibrational spectroscopy in phase transition identification. We have found that vibrational properties are very sensitive indicators to specify phases and phase transitions typical of unsaturated fatty acids at the molecular level. Using Raman spectroscopy we have identified high-temperature, middle-temperature and low-temperature phases of linoleic acid. Results obtained for linoleic acid were compared with parameters characteristic of α and γ phases of oleic acid - the parent compound of polyunsaturated fatty acids.