This paper studies, for the first time, the management of type information for an important class of semi-structured data: nested DAGs (Directed Acyclic Graphs) that describe execution traces of business processes (BPs for short). Specifically, we consider here type inference and type checking for queries over BP execution traces. The queries that we consider select portions of the traces that are of interest to the user; the types describe the possible shape of the execution traces in the input/output of the query. We formally define and characterize here three common classes of BP execution traces and their respective notions of type inference and type checking. We study the complexity of the two problems for query languages of varying expressive power and present efficient type inference/checking algorithms whenever possible. Our analysis offers a nearly complete picture of which combinations of trace classes and query features lead to PTIME algorithms and which to NP-complete or undecidable problems.