This bug was created in September 2008 in version 9.2 and it still persists. The Check Geometry tool often times will identify a polygon as as 'Self-Intersecting' which is great, but it fails to identify precisely where the self intersection occurs. Shown below a feature class of just two city footprints (out of 55 in the Salt Lake Valley). Notice that the Check Geometry tool reports that both polygons have self intersections. I'll add that Feature-ID 58 has a perimeter of 130 miles.
Running the Repair Geometry tool often times has unintended and undesirable results such as gaps, overlaps, and the elimination of once topologically consistent polygon edges. In my situation this can effect what jurisdiction/agency responds to a 9-1-1 call.
The initial bug submission requested adding feature to the Check Geometry tool that precisely locates the self intersection; that way one can edit just the problem manually instead of either living with the errors created by Repair Geometry or traversing miles and miles of feature(s) edges to make the edits. I think 8 years and several versions of ArcGIS is long enough for the user community to live with a known problem.