Measurements of neutrino spin are much more indirect than measurements of the spins of charged particles, simply because it's so rare to actually interact with the neutrinos. It's much more productive to examine the particles that you actually can detect and make inferences about the neutrinos. Here are some classic examples for you.
Neutrino discovery experiment
Pauli's original suggestion (1930) of the neutrino as a way to conserve energy and momentum in beta decay proposed that the neutrino would have to have spin-1/2, for statistical reasons. It took a while before Cowan and Reines (1956) observed the neutrino experimentally, in the inverse-beta reaction
$$ \rm \bar\nu_e + p \to n + e^+
$$
At this point the spin-1/2 character of the proton, neutron, and positron were all well-established, so some half-integer spin on the neutrino was required to conserve angular momentum. (Rescuing the continuous conservation laws was, after all, part of Pauli's motivation for proposing the neutrino in the first place.) I can't read the Cowan-Reines paper right now, but part of their success was correctly predicting the interaction cross-section for the process.
A spin-3/2 (or higher) neutrino would have required different assumptions about orbital angular momentum in the cross-section calculation, and the experimenters wouldn't have gotten that computation correct.
However, the Cowan-Reines experiment only observed that neutrinos had interacted with the detector, and didn't actually do anything to manipulate their angular momentum. So as measurements go, that's pretty indirect.
Muon spin measurement
Probably the first experimental manipulation of a spin that sheds light on the neutrino was the Garwin, Lederman, and Weinrich measurement (1957) of the muon spin made immediately after Wu's discovery of parity nonconservation. The Garwin et al. paper doesn't actually make a Stern-Gerlach-type measurement of the muon. Rather, the experiment looked at the two-step decay
\begin{align}
\pi^+ &\to \mu^+ + \nu_\mu
\\ \mu^+ &\to e^+ + 2\nu
\end{align}
The principal discoveries reported by Garwin et al. were that (a) muons emitted in pion decay are born polarized, which violates parity symmetry, and (b) the positrons emitted in the muon decay tend to be emitted along the muons' south poles, which also violates parity symmetry. These measurements were made by stopping the pion and muon parts of the cyclotron beam in different places, so that the muons were decaying at rest, and allowing the decay to proceed in a known magnetic field so that the muons' spins were allowed to precess. As a side effect, Garwin et al. got a measurement of the muons' spin precession rate in a given magnetic field, which they interpreted as a value for the muon magnetic moment and the muon $g$-factor. In a footnote they write that the $g$-factor is consistent with muon spin 1/2 and inconsistent with muon spin 3/2, referring to a computation of the $g$-factors in another paper.
The Garwin et al. paper doesn't discuss the neutrino spin, but (as in the Cowan-Reines reaction) you can get it indirectly from the pion decay,
$$ \pi^+ \to \mu^+ + \nu_\mu$$
The pion is a pseudoscalar particle, with spin zero. In order to conserve angular momentum, something has to balance the spin-1/2 apparently carried away by the muon; it's the neutrino.
I think that, like in the case of the Cowan-Reines experiment, there may have been some experimental wiggle room at this point for a spin-3/2 neutrino and something messy involving orbital angular momentum, but (also like the Cowan-Reines experiment) I think that would have predicted the wrong decay rates for the pion and muon.
The muon case is also interesting for your question since its spin was measured based on its magnetic properties but not with a Stern-Gerlach type measurement.
Goldhaber's "direct" neutrino helicity measurement
If you ask most folks about early measurements of neutrino spins, they'll point to the Goldhaber et al. observation (1958) that matter neutrinos have negative helicity. Goldhaber looked at the reaction chain
\begin{align}
\mathrm e^- + \mathrm{Eu}^*_{J=0} &\to \mathrm{Sm}^*_{J=1} + \nu
\\
\mathrm{Sm}^*_{J=1} &
\to \mathrm{Sm}_{J=0} + \gamma
% + \nu
\end{align}
The sample of radioactive europium, which is spinless, decays by electron capture to an excited state of samarium which is not spinless; that excited state goes back to a spinless ground state by emitting one photon.
If the second reaction happens soon enough that you can ignore interactions with the material (which is the case here)
and the neutrino and the photon are emitted back-to-back (which happens sometimes) then the photon and the neutrino must carry opposite spins.
But you can measure the polarization of gamma rays by measuring their transmission through magnetized iron.
There's actually an enormous amount of cleverness happening in the Goldhaber setup; not only was it necessary to infer a lot of details about the spins of the different states, but a coincidence in energy levels was required as well. Here's a nice blog writeup (2005).
The takeaway for you, from this long answer, should be that while the Stern-Gerlach effect was the first evidence for discrete spin, there are lots of other techniques that experimenters can use to get at the spin content of other systems. The literature is deep.
