When is an orbit spherical? I asked the following question over at math.stackexchange, but got no answers. Maybe it's less well-known than I thought, but I still wanted to ask here:
Let's assume we have an affine, reductive, algebraic group $G$ acting algebraically on a variety $X$, everything over an algebraically closed field of characteristic zero. Let $x\in X$ be some point with reductive stabilizer $H:=G_x$. Under what conditions on $x$ or $H$ is the orbit $G.x\cong G\newcommand{\qq}{/\hspace{-.8ex}/}\qq H$ a spherical variety? Let me briefly recall that a spherical variety is a homogeneous space $G\qq H$ satisfying one of the following, equivalent properties:


*

*Any Borel subgroup $B\subseteq G$ has an open orbit in $G\qq H$.

*Every equivariant completion of $G\qq H$ contains only finitely many orbits.

*For every irreducible $G$-module $V$ and any character $\chi$ of $H$,
$$\dim\left\{~v\in V \mid \forall h\in H: h.v = \chi(h)v ~\right\}\le 1.$$


I was hoping that this is well-known, but I cannot find any direct statements of that kind. Searching for the keywords "orbit" and "spherical" is quite fruitless because of property 1. 
Edit: In the cases of interest to me, the orbit $G.x$ is affine. 
 A: This was an active problem 20 years ago.  I don't know if it's completely resolved, but more can be said for affine spherical homogeneous spaces.  (Also---in general, a spherical variety need not be homogeneous, so searching for "spherical homogeneous variety might yield better results.)  Try Brion, "Classification des espaces homogènes sphériques" (Compositio, 1987) and more recent work of Knop et al, e.g., http://arxiv.org/abs/math/0505102 .  The latter has convenient tables at the end.
A: An extended comment:  As Dave points out, Michel Brion has been active in the study of spherical varieties (and some generalizations) in the setting of reductive algebraic groups in characteristic 0.    His work spans by now several decades, up to the present, and includes research papers in both French and English along with numerous surveys and lectures.    His classification result in the affine case (i.e., $H$ affine) was already outlined in his note:
Classification des espaces homogenes spheriques.
C. R. Acad. Sci. Paris S´er. I Math. 301 (1985), no. 18, 813–815.
A number of these papers are published in Comment. Math. Helv., while an influential joint paper with Luna and Vust can be found online at GDZ:
Espaces homogenes spheriques.
Invent. Math. 84 (1986), no. 3, 617–632.
I'm not sure whether this extensive work, or that of Knop et al. linked by Dave, will have all the information you want, but the subject remains active.
