This paper studies frames in Hilbert spaces generated by the orbits of (in)-finitely many vectors under a single operator, presenting new results on multiplication operators and operators composed of Jordan blocks, which generalizes existing results of Cabrelli, Molter, Paternostro and Philipp by means of techniques which deal with weighted interpolation, weighted composition operators, and Beurling--Lax theory related to shifts of infinity multiplicity. Likewise, we discuss Carleson frames and give counterexamples to a recent conjecture of Aldroubi, Cabrelli, Krishtal and Molter.