The ubiquitous nature of magnetohydrodynamic (MHD) waves and oscillations resolved in magnetic structures in the solar atmosphere is beginning to be documented. To date, the majority of the waves predicted by theory, i.e. the theory of (MHD) waves in a cylindrical, magnetic waveguide, have been observed in the solar corona. These oscillatory events appear to be driven by continuous motions present in the lower solar atmosphere, e.g. p-modes, granular buffeting, vortex motions. The amount of energy available from these drivers is extremely large and could make up a significant portion of the coronal heating budget. Further, in recent years, waves in the solar atmosphere have also begun to be exploited as magneto-seismological tools to measure plasma parameters that are hard to measure using direct methods. This makes study of the waves in the solar atmosphere of vital importance. Here we investigate the influence of geometric and dynamical effects on the eigenfrequencies and eigenfunctions of MHD oscillations supported by magnetic waveguides in solar atmospheric plasmas. The concept of magnetic waveguides is ideal for modelling the waves and oscillations that are observed in magnetic solar structures. Previous modelling efforts have studied simplified solar plasma waveguides but have provided a number of useful, initial magneto-seismological tools. We aim to generalise, further develop and extend these models to include more realistic and applicable loop geometries, i.e. elliptical loop cross-section and density stratification in an elliptically curved loop. We study how the stage of emergency is relevant for loop oscillations. In addition to this we provide the crucial first investigations into how dynamic plasma behaviour in the magnetic waveguides influences the oscillations. As coronal loops are observed to be cooling due to radiation, we investigate the influence of the radiative cooling of the background plasma on the oscillations. This latter topic is a very important and widely applicable development of MHD wave theory. It is found that if we are to obtain more accurate estimates of plasma parameters from the current cohort of magneto-seismological tools, then taking into account seemingly second order effects, e.g geometry and dynamism, will be more than essential. The second order, geometry (in terms of mathematics) effects are shown to cause important, measurable deviations from the first order theory and ignoring them can lead to substantial errors in the estimates of the plasma quantities (e.g. coronal magnetic field strength). With regards to dynamic plasmas, we find a number of very interesting and important results. The first is that the radiative cooling of a plasma also provides another vital and competing method for the damping of oscillations within the plasma (in addition to the previously suggested methods, e.g. resonant absorption, thermal conduction, etc.). The damping can be attributed to changes in temperature and density of the plasma. The majority of the 1 MK plasma in coronal loops is observed to be radiatively cooling, so the damping due to the cooling process could be a dominant method of wave damping in the solar atmosphere. Secondly, we also demonstrate three new magnetoseismological tools that can be used to measure cooling timescales in the corona. The inclusion of the plasma dynamism will also be necessary when calculating the contribution from waves to the heating of the coronal plasma. This novel investigation in to dynamic plasmas can also be applied to wave studies in a wide range of astrophysical plasma scenarios, as all plasmas experience some level of radiative cooling.