It is most prominent for myons in cosmic radiation or in particle accelerators, since these particles move nearly with the speed of light.
But even on intercontinental flights time dilation could be observed. Meanwhile a classic is the 1971 experiment by Hafele and Keating, who compared a clock resting on ground with an on-board clock.
And finally experiments are performed, in which atoms or ions are taken as clocks. Such an experiment was for instance performed in Heidelberg.