In light of recent observational results indicating an apparent lack of correlation between the Anomalous Microwave Emission (AME) and mid-infrared emission from polycyclic aromatic hydrocarbons (PAHs), we assess whether rotational emission from spinning silicate and/or iron nanoparticles could account for the observed AME without violating observational constraints on interstellar abundances, ultraviolet extinction, and infrared emission. By modifying the SpDust code to compute the rotational emission from these grains, we find that nanosilicate grains could account for the entirety of the observed AME, whereas iron grains could be responsible for only a fraction, even for extreme assumptions on the amount of interstellar iron concentrated in ultrasmall iron nanoparticles. Given the added complexity of contributions from multiple grain populations to the total spinning dust emission, as well as existing uncertainties due to the poorly-constrained grain size, charge, and dipole moment distributions, we discuss generic, carrier-independent predictions of spinning dust theory and observational tests that could help identify the AME carrier(s).