Is spherical trigonometry a dead research area? When I was an undergrad, the field of spherical trigonometry was cited as a once-popular area of math that has since died. Is this true? Are the results from spherical trigonometry relevant for contemporary research?
 A: It is not. As a proof, I will mention three relatively recent papers where I am a co-author:
M. Bonk and A. Eremenko, Covering properties of meromorphic functions, negative curvature and spherical geometry, Ann of Math. 152 (2000), 551-592.
A. Eremenko, Metrics of positive curvature with conic singularities on the sphere, Proc. AMS, 132 (2004), 11, 3349--3355.
A. Eremenko and A. Gabrielov, The space of Schwarz--Klein spherical triangles, Journal of Mathematical Physics, Analysis and Geometry, 16, 3 (2020) 263-282.
As you see, they are all published in mainstream math journals. All contain some new results on spherical triangles.
And I am not the only person who is involved in this business:
Feng Luo, A characterization of spherical polyhedral surfaces,
J. Differential Geom. 74(3): 407-424.
Edit. To address one comment: here is a forthcoming conference on spherical geometry
A: Not sure that it genuinely counts, but some interesting research in dynamical systems considers the dynamics of billiards of various shapes on the sphere. Examples:
Spina, M. E., & Saraceno, M. (2001). Quantum spectra of triangular billiards on the sphere. Journal of Physics A: Mathematical and General, 34(12), 2549.
Spina, M. E., & Saraceno, M. (1999). On the classical dynamics of billiards on the sphere. Journal of Physics A: Mathematical and General, 32(44), 7803.
Blumen, V., Kim, K. Y., Nance, J., & Zharnitsky, V. (2012). Three-period orbits in billiards on the surfaces of constant curvature. International Mathematics Research Notices, 2012(21), 5014-5024.
The reason I'm not sure is that the current research in classical, flat billiards would then entail that euclidean geometry is also not dead as a research field... but hey, I'm going to stand by that.
A: Other answers give examples of advanced research, but I thought I'd point out a modest result from my somewhat recreational investigations in the area:

If $W$ is area of the "hypotenuse-face" of a spherical tetrahedron with right "leg-faces" of area $X$, $Y$, $Z$, then
$$\cos\tfrac12W = \cos\tfrac12X\cos\tfrac12Y\cos\tfrac12Z+\sin\tfrac12X\sin\tfrac12Y\sin\tfrac12Z \tag{$\star$}$$

This is a counterpart of de Gua's theorem (aka, the "Pythagorean Theorem for Euclidean tetrahedra"):
$$W^2 = X^2 + Y^2+Z^2$$
Of course, a hyperbolic counterpart exists as well; just append "h"s to all the trig functions in $(\star)$, and change "$+$" to "$-$". There are even associated Laws of Cosines, but I won't go into that here.
I will give a shout-out to an ancient, still-open question of mine: "Pythagorean theorem for right-corner hyperbolic simplices?
", which specifically seeks the $4$-dimensional hyperbolic counterpart of $(\star)$. (Since spherical and hyperbolic relations are likely comparable, this also counts as an open question in spherical geometry.)
A: There is a lively site which publishes research questions, some of which essentially involve spherical geometry and trigonometry. Recent examples include:

*

*Cutting a spherical surface into mutually non-congruent pieces of
equal area

*Can you perturb an inscribed polytope so all its edges grow?
A: I’d say spherical trigonometry has never been an area of research mathematics. Instead, it is an area of mathematical techniques that have been useful for 1) other areas of mathematical research, and more importantly for 2) astronomy or 3) surveying and navigation. But those mathematical techniques are less known and less taught now than in the past, for good reason.

*

*The other answers here show that spherical trigonometry can be useful in convex geometry and ergodic theory and elsewhere in mathematical research. But spherical trigonometry is not itself the area of research — e.g. in the 2010 MathSciNet classification, the word “trigonometry” appears only in “97G60, plane and spherical trigonometry (educational aspects)”, under the top-level category of “97-XX, mathematics education”.


*Spherical trigonometry originated in astronomy, but much of its use there has been replaced with a more vectorial approach. E.g.: How would you calculate the angular distance $\theta$ between two celestial objects with given azimuths $a,a’$ and zenith distances $z,z’$? (Azimuth = compass direction of the closest point on the horizon.)
The traditional answer uses the spherical law of cosines: $$\cos \theta = \cos z \cos z’ + \sin z \sin z’ \cos(a-a’),$$ which is efficient for calculating with slide rules and trig tables. Most of us here would find it easier to compute unit vectors, take a dot product, and take the $\arccos$ of that. The calculations may be equivalent, but we don’t refer to the law of cosines in the process.


*Surveying and navigation used to be more important, and spherical trigonometry can be useful for both. Nowadays, we have bigger ships and require fewer navigators; we have aerial maps and require fewer surveyors; we have GPS and automated directions, and avoid these topics even more. This decline of surveying and navigation is the reason we teach less spherical trigonometry, which is why fewer mathematicians know it.
