We perform a detailed analysis of the band structure, phonon dispersion, and electron–phonon coupling of three types of small-radius carbon nanotubes (CNTs): (5,0), (6,0), and (5,5) with diameters 3.9, 4.7, and 6.8 Å respectively. The large curvature of the (5,0) CNTs makes them metallic with a large density of states at the Fermi energy. The density of states is also strongly enhanced for the (6,0) CNTs compared to the results obtained from the zone-folding method. For the (5,5) CNTs the electron–phonon interaction is dominated by the in-plane optical phonons, while for the ultrasmall (5,0) and (6,0) CNTs the main coupling is to the out-of-plane optical phonon modes. We calculate electron–phonon interaction strengths for all three types of CNTs and analyze possible instabilities toward superconducting and charge–density wave phases. For the smallest (5,0) nanotube, in the mean-field approximation and neglecting Coulomb interactions, we find that the charge–density wave transition temperature greatly exceeds the superconducting one. When we include a realistic model of the Coulomb interaction we find that the charge–density wave is suppressed to very low temperatures, making superconductivity dominant with the mean-field transition temperature around one K. For the (6,0) nanotube the charge–density wave dominates even with the inclusion of Coulomb interactions and we find the mean-field transition temperature to be around five Kelvin. We find that the larger radius (5,5) nanotube is stable against superconducting and charge–density wave orders at all realistic temperatures.