The subject has a lot of atention among specialists and the number of tools available depends on the amount of work you wnat to put on it. The paradox is that you want to avoid wet areas while stormwater engineers are looking for recreational areas to use them as ponds and damping peak flows.
There are three levels at which you can take your analysis:
Mapping of low points, also known as bluespots mapping
Mapping of runoff
Mapping of runoff with the stormwater management system
Bluespots are easy found by identifying depression in a DTM, under QGIS you have many alternatives, I would use the GRASS command r.fill.dir from the Processing Toolbox, but there are more tools to do it.
I have just found a nice video from Danmark showing how to do the bluespot mapping with other QGIS tools: Point Cloud to Bluespot, Stream and Drainage Basin (Catchment Area)
I cannot resits to comment that there are a couple of projects to address the more complex calculations of the runoff and runoff with pipes problem, but they are probably far from your needs and avalable resources:
- The project Itzí to create flood risk maps
- The just born project hex-utils-grass that could be use in flood modelling with hexagons (interesting videos about it:https://www.youtube.com/watch?v=uLO4HDCVBp0 and https://youtu.be/4lfuNHmsWLg)
- The Anuga project which has been used to simulate raifall on a 2d grid also