Language models (LMs) are capable of acquiring elements of human-like syntactic knowledge. Targeted syntactic evaluation tests have been employed to measure how well they form generalizations about syntactic phenomena in high-resource languages such as English. However, we still lack a thorough understanding of LMs’ capacity for syntactic generalizations in low-resource languages, which are responsible for much of the diversity of syntactic patterns worldwide. In this study, we develop targeted syntactic evaluation tests for three low-resource languages (Basque, Hindi, and Swahili) and use them to evaluate five families of open-access multilingual Transformer LMs. We find that some syntactic tasks prove relatively easy for LMs while others (agreement in sentences containing indirect objects in Basque, agreement across a prepositional phrase in Swahili) are challenging. We additionally uncover issues with publicly available Transformers, including a bias toward the habitual aspect in Hindi in multilingual BERT and underperformance compared to similar-sized models in XGLM-4.5B.