One objective of the 5G communication system and beyond is to support massive machine type of communication (mMTC) to propel the fast growth of diverse Internet of Things use cases. The mMTC is aimed at providing connectivity to tens of billions of sensor nodes. The dramatic increase of sensor devices and massive connectivity impose critical challenges for the network to handle the enormous control signaling overhead with limited radio resources. Nonorthogonal multiple access (NOMA) is a new paradigm shift in the design of multiple user detection and multiple access. NOMA with compressive sensing-based multiuser detection is one of the promising candidates to address the challenges of mMTC. The survey article is aimed at providing an overview of the current state-of-art research work in various compressive sensing-based techniques that enable NOMA. We present characteristics of different algorithms and compare their pros and cons, thereby providing useful insights for researchers to make further contributions in NOMA using compressive sensing techniques.