I am currently working on taxi-way route optimization. Are there any systems that store the taxi routes taken by the aircraft either by hour or by day or by airplane?
Your best bet will be to listen to recordings of taxi clearances issued by Ground Control.
The chance that you can see replays of enough ASDE-X or other surface radar recordings to get a meaningful data set is probably about zero unless you can get agreements with whatever ATC facility you're interested in.
While it's possible that some facility may record the data that you're asking about, the general answer is that the ground controller simply issues the clearance over the radio and the aircraft complies. If a conflict arises that requires something different, then the controller issues a revised clearance & the aircraft follows the new instruction. But unless a particular facility is doing something special (an extra person in the tower cab recording clearances & routings in order to undertake some specific analysis, for instance, or maybe a facility doing an analysis of ASDE-X tapes for some project), there's no widespread banking of that sort of thing.
Thus the suggestion that you will need to build your database yourself by listening to something like Live ATC (dot net) and copying down the routings given.