For the classical Stirling numbers of the second kind, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the Chebyshev–Stirling numbers, a special case of the Jacobi–Stirling numbers. Essential features are uniformity properties and the fact that the leading terms of the asymptotics are given explicitly and they contain elementary expressions only. Thereby supplements of the asymptotic analysis of these numbers are established.