I live in the north of England (where it's fairly wet) and have a section of lawn near a fence that is quite mossy and boggy.
I cannot put any drainage pipe in, because the only place to run the water would be into the next door neighbour's driveway ... the run-off water already heads that way but to put in deliberate pipes would not be a good idea.
It has been suggested that I plant a line of shrubs or bushes along the fence line to help dry out the soil.
I have talked to the local gardening stores (who were not very informative) and searched on Google (but I don't know the right search terms).
What I am looking for is a plant that grows no more than 6ft high, preferably 3-4ft and likes water. The current suggestion is a large number of different colored heather (which so far is looking possible).
Update: The site is along a fence line. Width is anything up to 18in or so. It gets the sun from early morning till late afternoon. Not sure about the soil acidity at the moment. It can be built up if needed. I would classify the soil as very soft and damp for most of the year.
Update #2: Width could be wider if I can get a better looking border. The reason the lawn is wet is that any rain (and we tend to get a reasonable amount) falls on the patch of grass outside my fence (public land) which then flows down onto my lawn and then off to the neighbour's. My lawn is flatter and hence the water pools there. The sewers and fresh water supply pipes are far enough away from the wet patches and should be fine, I have recently had the plans from the council detailing where they are. The area does begin to dry out if we have several hot days in a row, so I think that if we actually get a summer it might dry out completely.