Using experimental values of moduli of elasticity $C_{ij}$ and compliance moduli $S_{ij}$, the characteristic surfaces of Young’s moduli $E_{i}$ and dependences of the Poisson ratios $\mu_{ij}$($\varphi$, $heta$, $\psi$) for some single crystals of trigonal system are plotted. For the first time, the index auxeticity surfaces for NaAl$_{2}$O$_{3}$, Sb, Bi, CaCO$_{3}$, SbAs (25% As), Te, Se, $\alpha$-SiO$_{2}$, BaB$_{2}$O$_{4}$, As, and NaNO$_{3}$ single crystals are constructed. The criteria of appearance of the axial and non-axial auxeticity are found. Based on the analysis of the abnormal deformations of SiO$_{2}$ single crystal in the temperature range of 0–851 K, the mechanisms of origin and transformation of auxeticity are determined. As established, the $\alpha$-SiO$_{2}$ at T = 844 K is an absolute auxetic (the degree of auxeticity—$S_{ extrm{a}}$ = 1).