A series of inequalities involving Stirling numbers of the first and second kind with adjacent indices are obtained, some of which show log- concavity of Stirling numbers in three directions. Some of them are new, others extend or improve earlier results. These inequalities are used to prove unimodality or strong unimodality of all the subfamilies of Stirling probability functions for any parameter value. They are further applied to obtain Poisson distributions stochastically larger than some of the subfamilies of Stirling probability distributions, to show monotonicity of the convolutions of Stirling numbers and to prove log-concavity of the binomial coefficients.