You can use Select by Location tool. http://desktop.arcgis.com/en/arcmap/10.3/map/working-with-layers/using-select-by-location.htm
find features that don't intersect (by select features that do intersect, then grab all features not selected)
Since you are working with one layer try using the Intersect Analysis Tool
If you are looking to find isolated road segments, such as an "island" disconnected from the rest of network, you can use a single-source shortest path algorithm to find the distance of all network nodes to any given node on the main network. In ArcGIS, this can be done using the service area operation in Network Analyst, by using a large cut-off distance/time.
The isolated road segments will be outside of the service area. This works for the more general case when a small number of isolated segments are connected to each other but not to the main road network.