I have several shapesfiles which perfectly overlay but the coordinates reference system is unknown. I need to overly these with other data in a different coordinate system. I tried initially to overlay with several projected and geographic coordinates systems associated with the area out of which two showed very close displacement (about 60m). Next i tried different datum with one of the 'promising' coordinates reference systems (a geographic coordinates system) and also changed the units but same results. Attached is a clip of one of the coordinate extents. I am yet to try the second 'promising' CRS but decided to put it out just in case i am missing something
In such cases I usually:
Check X (Easting) and Y (Northing) coordinate values, they often indicate or provide some clues on which CRS they might be in. E.g. UTM X (Easting) coordinates will be typically 6 digits between +/- 200k and 700k, while Y (Northing) will depend on latitude (as they indicate distance from the Equator). Note also some local / national specific styles (e.g. adding UTM zone prefix to Easting in some German CRS).
Check with Spatial Reference or similar source for all potential CRS for a given area.
Prepare a sample reference dataset (e.g. boundaries, roads, rivers etc. depending on data) in a known coordinate system (e.g. WGS 84 - EPSG:4326).
Try to assign different CRS definitions to my data and overlay with reference dataset. Note: pay attention to datum transformations, the displacement of 60 meters can easily be caused by wrong set of transformation parameters.
and finally (best option but often not available) - contact your data providers :)