Separation logic is a powerful tool for reasoning about structured, imperative programs that manipulate pointers. However, its application to unstructured, lower-level languages such as assembly language or machine code remains challenging. In this paper we describe a separation logic tailored for this purpose that we have applied to x86 machine-code programs. The logic is built from an assertion logic on machine states over which we construct a specification logic that encapsulates uses of frames and step indexing. The traditional notion of Hoare triple is not applicable directly to unstructured machine code, where code and data are mixed together and programs do not in general run to completion, so instead we adopt a continuation-passing style of specification with preconditions alone. Nevertheless, the range of primitives provided by the specification logic, which include a higher-order frame connective, a novel read-only frame connective, and a ‘later’ modality, support the definition of derived forms to support structured-programming-style reasoning for common cases, in which standard rules for Hoare triples are derived as lemmas. Furthermore, our encoding of scoped assembly-language labels lets us give definitions and proof rules for powerful assembly language ‘macros’ such as while loops, conditionals and procedures. We have applied the framework to a model of sequential x86 machine code built entirely within the Coq proof assistant, including tactic support based on computational reflection.