In addition to the examples mentioned in the question, of manifolds and schemes, other commonly occuring types of locally ringed spaces are formal schemes and complex analytic spaces.
I don't know how extensive the taxonomy of locally ringed spaces is. For example, if $A$ is a local ring, we can form the locally ringed space consisting of a single point, with $A$ sitting on top of it. These are the topologically simplest locally ringed spaces (after the empty space). If $A$ is a field, one obtains a scheme. If $A$ is a complete local ring, one obtains a formal scheme. In general, this doesn't fit into any particular taxonomic grouping that I know of.
Incidentally, it might be worth mentioning that the various taxonomic classes can interact: for example, analytification of schemes over ${\mathbb C}$ is conveniently described in terms of maps (in the category of locally ringed spaces) to complex analytic spaces.
