If your goal is the find the most-used road segments, Dan's suggestion to the the OD Cost Matrix solver will not work. OD Cost Matrix does not return the used road segments but rather only gives you the travel time or distance between the points.
Instead, you need to use the Network Analyst Route solver. You need to set up your input Stops carefully so you can create one route between each start and end location. You can do this using the Route Name field. You need to map a unique identifier for each start-end location pair into the Route Name field. This will cause the Route solver to generate one route per unique Route Name, rather than putting all the points into the same route, which isn't what you want.
Once you have solved the route, you can use the Copy Traversed Source Features tool to see which road segments were actually traversed, and then you'll probably have to do some further processing to figure out the number of times each segment was traversed.
