Totally disconnected orbit spaces Let $X$ be a totally disconnected $G$-space, where $G$ is a locally compact Hausdorff group. Is the orbit space X/G also totally disconnected?

The same question for locally compact, Hausdorff, totally disconnected topological group $G$.
The orbit space is not necessarily disconnected. For a simple example, let $X$ be the
irrational reals, with the order topology and let $G$ be the rationals with the
discrete topology. $G$ is then locally compact, metrizable and totally disconnected, and it acts continuously on $X$ by the addition of real numbers.
It is easy to see though, that every orbit is dense in $X$, therefore the
orbit space is indiscrete.

Some additional notes:


Any subgroup of $\text{Homeo}(X)$ with the discrete topology has all
the required properties, including being totally disconnected.
Adding the additional requirement that $G$ is not discrete makes no 
difference, because you can e.g. multiply $X$ and $G$ by a
disconnected local field and let that act on itself by addition. This
produces essentially the same orbit space.
Requiring that the group is totally disconnected makes no real difference
anyway, since a continuous map into a totally disconnected space must be
constant on on every connected subset of its domain. In this case that
means that the action of the connected component of $G$ is always trivial.
Well, let's see. To show that $X/G$ is totally disconnected, (by which I assume you mean that every point is open), we need to show that every $y \in X/G$ is open.

Now, the open sets in a quotient space are by definition, the images under the projection map of the open sets of $X$. So, a set $S$ is open, if it is $\pi(U)$ for some open set $U$ in $X$ where $\pi: X \rightarrow X/G$ is the projection.

However, if $X$ is totally disconnected, then every subset of $X$ is open. Hence, 
$$y = \pi(\pi^{-1}(y))$$
is open and thus $X/G$ is totally disconnected.

This is actually an example of the more general fact that the images of totally disconnected spaces under open maps are totally disconnected.