1960 POTUS election:  RNC chairman Morton challenged the results in 11 states.  Only Hawaii changed on a recount, to a Democrat.

Texas fraud:  yes, they found some -- however, is was not enough to close the wide gap between the winner and loser.

Illinois fraud:  there were also irregularities in Republican controlled areas to offset the irregularities found in areas controlled by Democrats.

Convictions for Fraud:  3 Chicago election workers.  All other cases did not result in conviction (650).

FYI:  In the 60's there were still many conservative Dixiecrats in the increasingly liberal Democratic party during it's idealogical switch.  As well as a liberal to conservative switch for the Republican party.  If you notice, the western half of the U.S. was completely red when Nixon ran and lost.  So, voter fraud was difficult to pigeon-hole to a particular party in the centrist 60's.