  Given a maximally symmetric $d$-dimensional background with isometry algebra
$\mathfrak{g}$, a symmetric and traceless rank-$s$ field $\phi_{a(s)}$
satisfying the massive Klein-Gordon equation furnishes a collection of massive
$\mathfrak{g}$-representations with spins $j\in \{0,1,\cdots,s\}$. In this
paper we construct the spin-$(s,j)$ projectors, which are operators that
isolate the part of $\phi_{a(s)}$ that furnishes the representation from this
collection carrying spin $j$. In the case of an (anti-)de Sitter ((A)dS$_d$)
background, we find that the poles of the projectors encode information about
(partially-)massless representations, in agreement with observations made
earlier in $d=3,4$. We then use these projectors to facilitate a systematic
derivation of two-derivative actions with a propagating massless spin-$s$ mode.
In addition to reproducing the massless spin-$s$ Fronsdal action, this analysis
generates new actions possessing higher-depth gauge symmetry. In (A)dS$_d$ we
also derive the action for a partially-massless spin-$s$ depth-$t$ field with
$1\leq t \leq s$. The latter utilises the minimum number of auxiliary fields,
and corresponds to the action originally proposed by Zinoviev after gauging
away all St\"{u}ckelberg fields. Some higher-derivative actions are also
presented, and in $d=3$ are used to construct (i) generalised higher-spin
Cotton tensors in (A)dS$_3$; and (ii) topologically-massive actions with
higher-depth gauge symmetry. Finally, in four-dimensional $\mathcal{N}=1$
Minkowski superspace, we provide closed-form expressions for the analogous
superprojectors.
