In this very nice paper, the authors consider (quasi)orthogonal polynomial sequences [\((Q)OPS\): see (i)] with respect to a regular moment functional \(\sigma\) [see (ii)]. (i) \(\{P_n\}_{n=0}^{\infty}\) polynomials in \(x\) with degree \(P_n=n,\;n\geq 0\) form a \(QOPS\) resp. \(OPS\) if \(\langle \sigma,P_nP_m\rangle = K_n\delta_{nm}\) with \(K_n\in{\mathcal R}\) resp. \(K_n\in{\mathcal R}\setminus\{0\}\). (ii) A moment functional \(\sigma\) is regular resp. positive definite, if the moments satisfy a Hamburger condition \(\bigtriangleup_n(\sigma)=det[\langle \sigma,x^{i+j}\rangle]_{i,j=0}^n\not= 0\) resp. \(\bigtriangleup_n(\sigma)>0\). The main result is a set of classifications for discrete classical \(OPS\), including the generalisation of a result due to \textit{W. Hahn} [Math. Z. 43, 101 (1937; Zbl 0017.20601)] For a given \(OPS\) \(\{P_n\}_{n=0}^{\infty}\) with respect to a regular moment functional \(\sigma\) and a given integer \(r\geq 1\), the following statements are proven to be equivalent: \(\{\bigtriangledown^r P_n(x)\}_{n=0}^{\infty}\) is a \(QOPS\), there exist \(r+1\) polynomials \(\{a_k(x)\}_{k=r}^{2r}\) with \(a_{2r}(x) \not\equiv 0\), degree \(a_k\leq k\;(r\leq k\leq 2r)\) and \(\bigtriangleup (a_k\sigma)=a_{k-1}\sigma\;(r+1\leq k\leq 2r)\), there exist a moment functional \(\tau\not= 0\) and \(r+1\) polynomials \(\{a_k(x)\}_{k=r}^{2r}\) with degree \(a_k\leq k\;(r\leq k\leq 2r)\) and \(\bigtriangledown^{2r-k}\tau=a_k(x)\sigma\;(r\leq k\leq 2r)\), \(\{P_n(x)\}_{n=0}^{\infty}\) is a discrete classical \(OPS\).