Most stuttering detection and classification research has viewed stuttering as a multi-class classification problem or a binary detection task for each dysfluency type; however, this does not match the nature of stuttering, in which one dysfluency seldom comes alone but rather co-occurs with others. This paper explores multi-language and cross-corpus end-to-end stuttering detection as a multi-label problem using a modified wav2vec 2.0 system with an attention-based classification head and multi-task learning. We evaluate the method using combinations of three datasets containing English and German stuttered speech, one containing speech modified by fluency shaping. The experimental results and an error analysis show that multi-label stuttering detection systems trained on cross-corpus and multi-language data achieve competitive results but performance on samples with multiple labels stays below overall detection results.