Summary: Let \(\mathrm{GL}(n) = \mathrm{GL}(n, \mathbb{C})\) denote the complex general linear group and let \(G \subset \mathrm{GL}(n)\) be one of the classical complex subgroups \(\mathrm{O}(n)\), \(\mathrm{SO}(n)\), and \(\mathrm{Sp}(2k)\) (in the case \(n = 2k\)). We take a finite dimensional polynomial \(\mathrm{GL}(n)\)-module \(W\) and consider the symmetric algebra \(S(W)\). Extending previous results for \(G = \mathrm{SL}(n)\), we develop a method for determining the Hilbert series \(H(S(W)^G, t)\) of the algebra of invariants \(S(W)^G\). Our method is based on simple algebraic computations and can be easily realized using popular software packages. Then we give many explicit examples for computing \(H(S(W)^G, t)\). As an application, we consider the question of regularity of the algebra \(S(W)^{\mathrm{O}(n)}\). For \(n=2\) and \(n=3\) we give a complete list of modules \(W\), so that if \(S(W)^{\mathrm{O}(n)}\) is regular then \(W\) is in this list. As a further application, we extend our method to compute also the Hilbert series of the algebras of invariants \(\Lambda(S^2V)^G\) and \(\Lambda(\Lambda^2 V)^G\), where \(V = \mathbb{C}^n\) denotes the standard \(\mathrm{GL}(n)\)-module.