Edge localized modes (ELMs) are magnetohydrodynamic (MHD) instabilities that occur in the high confinement regime (H-mode) of magnetically confined fusion plasmas. ELMs lead to sudden periodic releases of particles and stored energy on a millisecond time scale. These ELM crashes might cause intolerably high heat fluxes onto the divertor target plates or the first wall in future fusion devices. According to the broadly accepted linear peeling-ballooning model these MHD instabilities are driven by edge current density and steep edge pressure gradient, which are characteristic for the H-mode. However, details of the underlying process responsible for ELMs and their nonlinear development during the crashes are not yet fully understood. The focus of this thesis is to determine one of the main characterizing parameters of MHD instabilities, which is the periodic magnetic structure described by the poloidal and toroidal mode numbers m and n. These structures are investigated for ELMs and associated phenomena on the ASDEX Upgrade tokamak. Mode numbers of instabilities are determined by recently upgraded magnetic pick-up coil arrays. It is shown that mode numbers of high frequency oscillations, f > 50 kHz, can only be reliably determined if the frequency dependent phase response of the coils is taken into account. Furthermore, a precise ELM synchronization enables the identification of mode numbers during the fast crash of ELMs, which was never achieved before on ASDEX Upgrade. In addition to that, mode numbers and positions of modes appearing between ELM crashes as well as their connection to the edge gradient development are determined for the first time, which is a big step forward in characterizing them and understanding their role for the ELM itself. Ensembles of modes between ELM crashes are detected with different rotation velocities and thereby different locations at the plasma edge. Modes with higher toroidal mode numbers, n=7-13, appear at the position of fastest poloidal plasma rotation, close to the maximum pressure gradient and might be interpreted as ideal modes without additional phase velocity. Modes with lower toroidal mode numbers, n=2-7, exist further outwards close to the separatrix. A similar low n structure is present during the ELM crash. The detection of this structure and other parameters of the crash such as induced energy losses or duration enables a quantitative comparison to results from modeling with the nonlinear MHD code JOREK for the first time. Here the n=6 component with smaller structure size is linearly dominant, but nonlinear coupling in which n=1 is particularly important leads to the dominance of larger structure sizes with n=3-5 during the ELM crash which is in excellent agreement with experimental observations. Moreover, the scaling of the toroidal and poloidal structure, intensity and duration of the ELM crash with plasma parameters is investigated in a database containing various plasma scenarios. It is found that n increases linearly with the inclination of the magnetic field lines, i.e. decreases with safety factor q. Furthermore, no intense ELMs are found at high edge q and no long lasting ELMs are found at low edge q. Other parameters such as normalized pressure gradient alpha, bootstrap current density jBS or plasma triangularity delta, that should have, according to linear peeling-ballooning theory, an impact on n, do not show clear trends. Introducing a simple geometric model, the scaling of toroidal structure size with q can be explained by the dominance of one poloidal structure. In order to place the nonlinear phase of ELMs into a wider context of other nonlinear edge phenomena, toroidal mode numbers are analyzed between ELM crashes on the JET tokamak and during ELM crashes mitigated with an external magnetic error field, ELM crashes of nitrogen seeded discharges and ELM-like magnetic bursts of the intermediate confinement regime (I-phase). The JET edge modes are found with similar properties as the ones on ASDEX Upgrade. The ELM crash structure is found to adopt to the one of the external error field while nitrogen seeding seems not to change it. I-phase bursts have the same toroidal structure as ELMs.