Summary: Given a semigroup \(S\), for each Green’s relation \(\mathcal{K} \in \{ \mathcal{L}, \mathcal{R}, \mathcal{J}, \mathcal{H} \}\) on \(S\), the \(\mathcal{K} \)-height of \(S\), denoted by \(H_{\mathcal{K}}(S)\), is the height of the poset of \(\mathcal{K} \)-classes of \(S\). More precisely, if there is a finite bound on the sizes of chains of \(\mathcal{K} \)-classes of \(S\), then \(H_{\mathcal{K}}(S)\) is defined as the maximum size of such a chain; otherwise, we say that \(S\) has infinite \(\mathcal{K} \)-height. We discuss the relationships between these four \(\mathcal{K} \)-heights. The main results concern the class of stable semigroups, which includes all finite semigroups. In particular, we prove that a stable semigroup has finite \(\mathcal{L} \)-height if and only if it has finite \(\mathcal{R} \)-height if and only if it has finite \(\mathcal{J} \)-height. In fact, for a stable semigroup \(S\), if \(H_{\mathcal{L}}(S) = n\) then \(H_{\mathcal{R}}(S) \leq 2^n - 1\) and \(H_{\mathcal{J}}(S) \leq 2^n - 1\), and we exhibit a family of examples to prove that these bounds are sharp. Furthermore, we prove that if \(2 \leq H_{\mathcal{L}}(S) < \infty\) and \(2 \leq H_{\mathcal{R}}(S) < \infty \), then \(H_{\mathcal{J}}(S) \leq H_{\mathcal{L}}(S) + H_{\mathcal{R}}(S) - 2\). We also show that for each \(n \in \mathbb{N}\) there exists a semigroup \(S\) such that \(H_{\mathcal{L}}(S) = H_{\mathcal{R}}(S) = 2^n + n - 3\) and \(H_{\mathcal{J}}(S) = 2^{n + 1} - 4\). By way of contrast, we prove that for a regular semigroup the \(\mathcal{L}\)-, \(\mathcal{R} \)- and \(\mathcal{H} \)-heights coincide with each other, and are greater or equal to the \(\mathcal{J} \)-height. Moreover, in a stable, regular semigroup the \(\mathcal{L}\)-, \(\mathcal{R}\)-, \(\mathcal{H} \)- and \(\mathcal{J} \)-heights are all equal.