Let K be an infinite field and let \(W_1\) and \(U_1\) be the Lie algebras of the derivations of the algebras of polynomials K[t] and of Laurent polynomials \(K[t,t^{-1}]\) , respectively. The algebras \(W_1\) and \(U_1\) admit natural \(\mathbb {Z}\) -gradings. We provide a basis for the graded identities of \(W_1\) and of \(U_1\) and prove that they do not admit any finite basis. Additionally, we provide a basis for the identities of certain graded Lie algebras with a grading such that every homogeneous component has dimension \(\le 1\) , assuming that a basis of the multilinear graded identities is known. The \(\mathbb {Z}\) -graded identities for \(W_1\) , in characteristic 0, were described in the paper [10]; we give an alternate and characteristic-free proof of the main theorem of [10]. It should be noted that our results hold also in characteristic 2, although the proofs are rather different than those when the characteristic of K is different from 2. We also describe the graded identities of the Virasoro algebras assuming K is of characteristic different from 2 and 3. Furthermore, the methods developed here make it possible to describe bases of the graded identities for certain graded algebras equipped with a grading known as fine. For instance, the special linear Lie algebra \(sl_q(K)\) with the Pauli gradings, where q is a prime, can be considered.