Step 1: Understand the question.
The question asks to identify the correct statements regarding different types of flagella (flagellae) based on their structure.
Step 2: Evaluate each statement based on the known characteristics of flagella types.
\begin{itemize}
\item Statement I: Stichonematic flagellae possess a single row of lateral appendages.
Stichonematic flagella are characterized by the presence of a single row of fine, hair-like lateral appendages (also known as mastigonemes or flimmer hairs) on one side of the flagellum. This statement is Correct.
\item Statement II: Pantonematic flagellae are completely devoid of mastigonemes.
Pantonematic flagella are characterized by having two or more rows of mastigonemes (lateral appendages) distributed all around the flagellum. Flagella completely devoid of mastigonemes are typically called whiplash or acronematic flagella. Therefore, this statement is Incorrect.
\item Statement III: Acronematic flagellae have a naked terminal part of the axoneme.
Acronematic flagella (also known as whiplash flagella) are smooth, lacking lateral appendages. They often taper into a naked (uncovered by mastigonemes or extensions of the flagellar membrane) terminal filament, which is an extension of the axoneme. This statement is Correct.
\item Statement IV: Pantacronematic flagellae are completely devoid of terminal naked filament.
Pantacronematic flagella are a type of pantonematic flagella (having multiple rows of mastigonemes) that also possess a terminal naked filament, similar to acronematic flagella. The statement claims they are completely devoid of it, which is contrary to their definition. Therefore, this statement is Incorrect.
\end{itemize}
Step 3: Identify the correct option based on the evaluation.
Based on the analysis, statements I and III are correct, while statements II and IV are incorrect.
The option that includes both I and III is the correct one.
The final answer is $\boxed{\text{I and III}}$.