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We present and analyze low-temperature series and complex-temperaturepartition function zeros for the $q$-state Potts model with $q=4$ on thehoneycomb lattice and $q=3,4$ on the triangular lattice. A discussion is givenas to how the locations of the singularities obtained from the series analysiscorrelate with the complex-temperature phase boundary. Extending our earlierwork, we include a similar discussion for the Potts model with $q=3$ on thehoneycomb lattice and with $q=3,4$ on the kagom\'e lattice.Comment: 33 pages, Latex, 9 encapsulated postscript figures, J. Phys. A, in  pres

Study of the Potts model on the honeycomb and triangular lattices: Low-temperature series and partition function zeros