This paper studies some famous classes of polynomials that can be defined via classical combinatorial objects. The main results of the paper are congruences that these polynomials satisfy modulo a prime \(p\). For example, define the \(r\)-derangement polynomials \(\mathcal{D}_{n,r}(x)\) by \(\mathcal{D}_{n,r}(x)=\frac{1}{r!}\sum_{j=0}^n{\binom{n}{j}}(j+r)!(x-1)^{n-j}\). The authors show that \[ \mathcal{D}_{mp+q,r-1}(1-x)\equiv(-x)^{mp}\mathcal{D}_{q,r-1}(1-x)\pmod p. \] As another example, consider the \(n^{\text{th}}\) Lah polynomial \(\mathcal{L}_n(x)=\sum_{k=0}^nL(n,k)(x)_k\), where \(L(n,k)\) is the number of partitions of the set \([n]\) into \(k\) ordered lists and \[ (\alpha)_r=\alpha(\alpha-1)\cdots(\alpha-r+1). \] The authors show that \[ (-x)^{m+n-2}\sum_{k=0}^{p-1}(-m)_{p-1-k}\mathcal{L}_{n+k}(x)\equiv x^{p-1}\sum_{j=0}^n(-1)^jL(n,j)\mathcal{D}_{j+m+n-2}(1-x)\pmod p. \] The paper extends further, giving congruences involving the so-called \((r_1,\ldots,r_q)\)-Bell polynomials.