Aims:We combine the magnetohydrodynamic (MHD) theory of resonantly damped quasi-mode kink oscillations with observational estimates of the period and damping of transverse coronal loop oscillations to extract information on physical parameters in oscillating loops. Methods: A numerical study of the quasi-mode period and damping, in one-dimensional fully non-uniform flux tubes, is used to obtain equilibrium models that reproduce the observed periods and damping rates. This scheme is applied to 11 loop oscillation events. Results: When only the damping rate is used, the valid equilibrium models form a one-dimensional solution curve in the two-dimensional parameter space (density contrast, transverse inhomogeneity length-scale). Lower limits to the transverse inhomogeneity are obtained in the limit of high contrast loops. When both the period and the damping rate are used, the equilibrium Alfvén speed (or Alfvén travel time) comes into play. The valid equilibrium models then form a one-dimensional solution curve in the three-dimensional parameter space (density contrast, transverse inhomogeneity length-scale, Alfvén speed or Alfvén travel time). The projection of these solutions onto the Alfvén speed axis is found to be constrained to a rather limited interval. Upper limits to the internal Alfvén speed are derived for 9 of the 11 analysed events.