The detection of a property called "single-threading" in functional programs or denotational language definitions can be exploited to produce a more efficient implementation of the program or language, by allowing a program variable or semantic domain to be implemented by a global data structure. Earlier work by David A. Schmidt has given sufficient criteria for the detection of the single-threading property in [lambda]-calculus expressions. We extend that work by giving criteria for single-threading detection in combinator notations;Two sets of single-threading criteria are given: one for a particular combinator language, TML, and the other for a generalized combinator notation. In both cases, proofs are given which demonstrate the soundness of the criteria. We also discuss some reasons for the differences in the two sets of criteria;In order to evaluate the usefulness of single-threading detection and the associated globalization transformation in practice, a testbed implementation of the TML criteria is being programmed, using the PSI/DAOS compiler-generation system developed at Aalborg University. Some preliminary results from this work are discussed.