Surfactants are surface active agents that accumulate at fluid interfaces and influence interfacial properties, e.g. the surface tension. For single rising bubbles, even a small amount of surfactant causes Marangoni forces that influence the bubble rise significantly. In this work, Direct Numerical Simulations (DNS) with an Arbitrary Lagrangian-Eulerian (ALE) Interface-Tracking method are performed. The use of a subgrid-scale model enables the simulation of realistic time and length scales and the comparison with experiments. The resolution requirements close to the interface are examined using 2D simulations to reduce the computational costs further. Then, 3D simulations of single rising bubbles under the influence of Triton-X100 are carried out, investigating different bubble diameters and initial surfactant bulk concentrations. The 3D simulations provide new insights into the transition from a helical motion into a zig-zag motion, which can only be observed in the presence of a surfactant. Additionally, the reciprocal influence of the local surfactant distribution on the interface and the vortex structures for path-unstable bubbles are analysed. Finally, the local surfactant distribution on the interface is modelled using a data-driven approach. The model is based on the DNS data obtained from the 3D simulations and is in good agreement with the validation data. In future work, the derived model can be used to improve existing simplified models for the simulation of bubbly flows under the influence of surfactant.