The Rim-to-Rim (R2R), R2R2R, and R2R2R2R2R (! aka 2xR2R2R) have been huge classics for many years. A speed record here really counts as a world-class performance, as many of the best mountain runners have tested their mettle in the Canyon. The fastest times are typically set by running the North and South Kaibab Trails.
Note that the R2R is typically run N to S, since the North Rim is about 1000' higher than the south. However, some FKTs have been set as part of the R2R2R, and therefore in the other direction. The R2R2R has basically always been run starting and finishing on the South Rim (Yaki Point). This is because the North Rim is usually closed (inaccessible to vehicles) during times of the year when the weather is favorable (cool enough) to do this.
Davy Crockett has an excellent, detailed historical look at the history of Grand Canyon crossings on the Kaibab and Bright Angel trails, complete with many cool photos.
The R2R2R via the South & North Bass trails has its own Route Page here: R2R2R-alt
*** Belt Buckle Route! ***
Rickey Gates and partner Elizabeth Thorp have created a series of custom belt buckles commemorating some of the most iconic and hotly contested FKT routes! The R2R2R certainly qualifies. Each buckle is awarded to the FKT holder, and passed on when that FKT is superseded. Currently this buckle is held by Jim Walmsley for the men and Cat Bradley for the women.