Ordinals and ordinal notation systems play an important role in program verification, since they can be used to prove termination of programs. We present three ordinal notation systems representing ordinals below $\varepsilon_0$ in type theory, using recent type-theoretical innovations such as mutual inductive-inductive definitions and higher inductive types. As case studies, we show how basic ordinal arithmetic can be developed for these systems, and how they admit a transfinite induction principle. We prove that all three notation systems are equivalent, so that we can transport results between them using the univalence principle. All our constructions have been implemented in cubical Agda.