There are lots of overlaps between small world and scale-free, but I think much less so between those two and expanders.
The terms "small world" and "scale-free" are often used informally, but formal definitions are often along the lines of:
Small-world means short average (or maximum) path length (typically $O(\log n)$, with $n$ vertices) and highly clustered (meaning that for any vertex $v$, the fraction of pairs of neighbors of $v$ which are adjacent is high)
Scale-free is often taken to mean that the degree distribution follows some variant of a power-law (e.g. power-law with cutoff, etc.), but more generally/informally is used to mean that the degree distribution is long-tailed, in contrast to, say, Erdos-Renyi random graphs which have a degree distribution that is exponentially concentrated around its mean.
Expander, of course, has a formal definition that is almost 100% standardized.
Expanders by their nature have logarithmic diameter, similar to small-world networks. Beyond that, however, as far as I know there is little overlap between the concepts.
In practice, expanders are often bounded-degree or even regular of bounded degree, whereas "real-world" graphs typically have a long-tailed degree distribution, with a small (but surprisingly large - e.g. not exponentially small) number of high-degree "hubs" (as they are usually called). Furthermore, scale-free graphs (almost by definition, depending on your definition) have a large number of vertices of very low degree - in most "real-world" graphs there are a large number of vertices of degree 1 or 2. This makes real-world graphs very unlike expanders, in that it is very easy to disconnect real-world graphs by removal of targeted edges, whereas expanders are by their nature highly connected. (Real-world graphs often also have the property that removal of random edges is very bad at disconnecting them.)
I'm sure there is something to be said about the use of spectral techniques for things like community detection, etc. in real-world networks, and from that viewpoint there may or may not be a little more overlap with expanders, but I'm not an expert in that area (maybe we can get Mark Newman to join cstheory.SE to comment...)
