= Solution
Use mostly-plus Lorentzian signature and $\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}$, with antisymmetrized gamma products of unit weight. A <Majorana spinor> obeys a reality condition under <charge conjugation>:
$$
\boxed{\psi^c=C\overline\psi^{\,T}=\psi,\qquad C\gamma^\mu C^{-1}=-(\gamma^\mu)^T.}
$$
A <Weyl spinor> instead has definite <chirality>,
$$
\boxed{P_\pm\psi=\psi,\qquad P_\pm=\tfrac12(1\pm\gamma_5).}
$$
A four-dimensional <Majorana spinor> has four real components, while a <Weyl spinor> has two complex components. <Charge conjugation> reverses <chirality> in this signature, so a nonzero spinor cannot satisfy both conditions simultaneously.
For the <Dirac equation> $(\not\partial+m)\psi=0$, the derivative term reverses <chirality> because $\gamma_5$ anticommutes with every <gamma matrix>, whereas the <mass> term preserves it. Projecting a purely chiral spinor equation onto its two chiralities therefore gives both $\not\partial\psi=0$ and $m\psi=0$. Thus \b[a nonzero four-component <Weyl spinor> satisfying this uncoupled <Dirac equation> must be massless]. A <Majorana spinor> contains conjugate opposite-chirality components and can have a nonzero <mass>. Its charge-conjugation reality condition identifies particle and antiparticle operators, so \b[a Majorana particle is its own antiparticle]. A Weyl field does not by itself impose that identification.
This does not forbid writing a <Majorana mass term> for a neutral two-component Weyl field: such a term couples the field to its conjugate, and the resulting four-component massive field is Majorana, not a purely chiral solution of the uncoupled <Dirac equation>.
For the <Rarita-Schwinger field>, set $\chi=\gamma^\mu\psi_\mu$ and $d=\partial^\mu\psi_\mu$. The antisymmetric kinetic operator is invariant under
$$
\delta\psi_\mu=\partial_\mu\eta,
$$
because commuting derivatives contract to zero against $\gamma^{\mu\nu\rho}$. Expanding its gamma product gives
$$
E^\mu=\gamma^{\mu\nu\rho}\partial_\nu\psi_\rho
=\not\partial\psi^\mu-\partial^\mu\chi+\gamma^\mu(\not\partial\chi-d).
$$
Gamma contraction gives $\gamma_\mu E^\mu=2(\not\partial\chi-d)$, so the equation implies $d=\not\partial\chi$ and $\not\partial\psi^\mu=\partial^\mu\chi$.
For a <Fourier mode> with <momentum> $k$, if $k^2\ne0$ the latter identity gives $k^2\psi^\mu=k^\mu\not k\chi$. The mode is proportional to $k^\mu$ and hence pure gauge. Every physical nonzero-momentum mode therefore has \b[$k^2=0$]. For a null mode, choose an index with $k^\mu\ne0$. The same equation makes $\chi$ an element of the image of $\not k$, allowing a gauge choice with $\chi=0$. The field equations then reduce to
$$
\boxed{\gamma\cdot\psi=0,\qquad\partial\cdot\psi=0,\qquad\not\partial\psi_\mu=0.}
$$
Residual <gauge transformations> obey $\not\partial\eta=0$.
Take a null <momentum> in the third spatial direction. Residual gauge freedom and transversality remove the temporal and longitudinal <vector> components, leaving two transverse <vector> polarizations. Their helicities are $\pm1$, while on-shell spinor helicities are $\pm1/2$. The gamma-trace constraint removes the two combinations of total <helicity> $\pm1/2$ and retains the aligned combinations of <helicity> $\pm3/2$. Explicitly, in a <chiral representation> the transverse gamma contractions use $\sigma_1\pm i\sigma_2$, which annihilate the correspondingly aligned <spin> states and act nontrivially on the opposite states. Thus a Majorana <Rarita-Schwinger field> has \b[two physical massless polarizations]. The Majorana assumption is the one relevant to simple <supergravity>; a complex vector-spinor has these helicities for its particle and independent antiparticle modes.
A consistent free massive deformation is the algebraic term
$$
\mathcal L_m=\frac m2\overline\psi_\mu\gamma^{\mu\nu}\psi_\nu
$$
added to a kinetic term $-\tfrac12\overline\psi_\mu\gamma^{\mu\nu\rho}\partial_\nu\psi_\rho$. Its equation is
$$
\gamma^{\mu\nu\rho}\partial_\nu\psi_\rho-m\gamma^{\mu\rho}\psi_\rho=0.
$$
For $m\ne0$, divergence and gamma contraction give, respectively,
$$
\gamma^{\mu\rho}\partial_\mu\psi_\rho=0,\qquad
2\gamma^{\nu\rho}\partial_\nu\psi_\rho-3m\gamma^\rho\psi_\rho=0.
$$
These imply the <massive Rarita-Schwinger constraints> $\chi=0$ and $d=0$, and the remaining equation is $(\not\partial+m)\psi_\mu=0$. Squaring gives $(\Box-m^2)\psi_\mu=0$ in the chosen signature. The <gauge symmetry> is lost. In the rest frame, transversality sets the temporal component to zero; three spatial <vector> components times two on-shell spinor states give six states, and gamma trace removes the two spin-one-half states. The result is \b[four massive spin-$3/2$ states], with <spin> projections $\pm3/2,\pm1/2$. In <supergravity> the additional longitudinal states can be supplied by the <super-Higgs mechanism>.
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