Which Open Source GIS application support Linear Referencing and Dynamic Segmentation?

The other option that I have found is using PostGIS. This software provides the spatial extensions for the PostgreSQL rdbms. These are both Open Source. There is a Linear Referencing Component described here: Link.

If you want to be able to access this functionality through a more full-featured GIS program, then you might use the Linear Referencing extension for QGIS.

You may load the plugin for QGIS through the Plugin Menu. Use the Fetch Python Plugins command, then filter the search like the screenshot below:

Where the button currently shows "Reinstall", you will have the option to simply "Install" the plugin.

These are all open source products, and have active development, so I would say they are definitely worth checking out.