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We examine thermal warm dark matter (WDM) models that are being probed by current constraints, and the relationship between the particle dark matter spin and commensurate thermal history. We find significant corrections to the linear matter power spectrum for given thermal WDM particle masses. Two primary classes are examined: spin-1/2 particles (e.g., thermalized sterile neutrinos, axinos) and thermal spin-3/2 particles (e.g., gravitinos or non-supersymmetric particles). We present new transfer function fits for thermal WDM candidates in particle mass regimes beyond the range of previous work, and at the scales of current and upcoming constraints. Importantly, we find that the standard, predominantly used, spin-1/2, thermal WDM particle produces a colder transfer function than that determined in previous work. We also analyze the entropy requirements for these WDM models to successfully produce observed dark matter densities. We explore the early Universe physics of gravitinos as either partially thermalized or fully thermalized species, which considerably changes the particle dark matter candidates' thermalization history and effects on structure formation. For the first time, we also calculate the transfer function for thermal spin-3/2 WDM.