  The last decade has seen many attempts to generalise the definition of modes,
or MAP estimators, of a probability distribution $\mu$ on a space $X$ to the
case that $\mu$ has no continuous Lebesgue density, and in particular to
infinite-dimensional Banach and Hilbert spaces $X$. This paper examines the
properties of and connections among these definitions. We construct a
systematic taxonomy -- or `periodic table' -- of modes that includes the
established notions as well as large hitherto-unexplored classes. We establish
implications between these definitions and provide counterexamples to
distinguish them. We also distinguish those definitions that are merely
`grammatically correct' from those that are `meaningful' in the sense of
satisfying certain `common-sense' axioms for a mode, among them the correct
handling of discrete measures and those with continuous Lebesgue densities.
However, despite there being 17 such `meaningful' definitions of mode, we show
that none of them satisfy the `merging property', under which the modes of
$\mu|_{A}$, $\mu|_{B}$ and $\mu|_{A \cup B}$ enjoy a straightforward
relationship for well-separated positive-mass events $A,B \subseteq X$.
