Based on Detlev's answer, I have developed a QGIS plugin which should allow others to solve similar problems easily. It is available in the official QGIS plugins repository and can be found in the QGIS menu: Plugins / Manage and Install Plugins... and search for Disconnected Islands.
http://plugins.qgis.org/plugins/disconnected-islands/
This plugin runs on a line layer, building up a road (or rail, etc.) network graph of connected links. It then analyses connected subgraphs, ones that are connected to each other, but not connected to isolated or floating links. It creates an additional attribute containing the group ID of the subgraph. This can then be used to style the layer with Categorised styles, or Zoom to selection. The disconnected links can then be fixed.
Sample data to test this plugin can be found in your plugins directory: ~/.qgis2/python/plugins/disconnected-islands/sample-data/islands.zip
Source code can be forked from: https://github.com/AfriGIS-South-Africa/disconnected-islands