The three ways of generalization of canonical coherent states are briefly reviewed and compared with the emphasis laid on the (minimum) uncertainty way. The characteristic uncertainty relations, which include the Schrödinger and Robertson inequalities, are extended to the case of several states. It is shown that the standard $SU(1,1)$ and $SU(2)$ coherent states are the unique states which minimize the second order characteristic inequality for the three generators. A set of states which minimize the Schrödinger inequality for the Hermitian components of the $su_q(1,1)$ ladder operator is also constructed. It is noted that the characteristic uncertainty relations can be written in the alternative complementary form.