In this work we consider the rich spectrum of fast MHD modes in low-beta, line-tied coronal loops (standing and leaky). One of the most important conclusions is that in a straight and homogeneous cylindrical flux tube there should be observational signatures of the excitation of higher order harmonics. Nevertheless, our results indicate that these modes become leaky with the addition of some structure along the loop and that leakage can be quite efficient in damping the oscillations. After that, we study the effects in the modes of the addition of density structure along the loop axis: using an isobaric and thermal equilibrium with different heating functions, a density profile for a loop is derived, and then its oscillatory modes are studied. The main result is that the frequency and spatial structure of the trapped modes are very sensitive to variations of density inside the loop, specially in the apex, which depends strongly on the footpoint density and the ratio of heating to thermal conduction. Moreover, different tested heating profiles give more or less the same results that for the case of constant heating all along the loop.