Large language models (LLMs) appear successful in emulating compositional language, yet it remains unclear what these results entail about their underlying compositional semantic representations. The probing classifier paradigm has emerged as a tool to remedy this. This paper proposes to critically review the findings of 24 probing studies targeting a wide range of linguistic and semantic phenomena. It proposes a taxonomy of probing tasks based on the linguistic primitives they presuppose, distinguishing four tiers: lexical semantics, the syntax–semantics interface, propositional semantics, and discourse and pragmatics. A gradient in representational evidence emerges: LLMs robustly encode lexical information, display less consistent sensitivity to structural relations within sentences, and obtain unsatisfactory results on tasks requiring propositional content, speech acts, or pragmatic inference. The review underscores the need for a clearer theoretical grounding of what probing tasks measure and reflects on how probing can illuminate the compositional pathways available within current language models.