When cabling a network using fibre, what is the difference between single-mode and multi-mode fibre? When should I be using one or the other? Are there compatibility and/or speed concerns with either?
Main difference:
Singlemode fiber has a lower power loss characteristic than multimode fiber, which means light can travel longer distances through it than it can through multimode fiber. Not surprising, the optics required to drive singlemode fiber are way more expensive, especially considering any varying circumstances. 

When to use each:
Both singlemode and modern multimode fiber can handle 10G speeds.  The most important thing to consider is the distance requirement.  Within a data center, it's typical to use multimode which can get you 300-400 meters.  If you have very long runs or are connecting over longer distance, single mode can get you 10km, 40km, 80km, and even farther - you just need to use the appropriate optic for the distance required, and again, the prices go up accordingly.

Compatibility issues:
They are not compatible.  You cannot mix multimode and singlemode fiber between two endpoints.  The optics are not compatible either.

There's a lot more to say about fiber, in general, but I hope this answers your immediate question.
