In this contribution, we compare basic neural networks with convolutional neural networks for cut failure classification during fiber laser cutting. The experiments are performed by cutting thin electrical sheets with a 500 W single-mode fiber laser while taking coaxial camera images for the classification. The quality is grouped in the categories good cut, cuts with burr formation and cut interruptions. Indeed, our results reveal that both cut failures can be detected with one system. Independent of the neural network design and size, a minimum classification accuracy of 92.8% is achieved, which could be increased with more complex networks to 95.8%. Thus, convolutional neural networks reveal a slight performance advantage over basic neural networks, which yet is accompanied by a higher calculation time, which nevertheless is still below 2 ms. In a separated examination, cut interruptions can be detected with much higher accuracy as compared to burr formation. Overall, the results reveal the possibility to detect burr formations and cut interruptions during laser cutting simultaneously with high accuracy, as being desirable for industrial applications.