Wibbly wobbly, timey wimey

"A big ball of wibbly wobbly, timey wimey stuff" was how the Tenth Doctor described time to Sally Sparrow through a DVD Easter egg. Though he quickly admitted that the sentence had "got away from [him]", the term was soon thereafter applied to an invention of his, the timey-wimey detector. (TV: Blink)

Though it was seemingly his tenth incarnation who coined it, the Fifth Doctor was familiar enough with the expression to complete the phrase when the Tenth Doctor started it. (TV: Time Crash) According to one account, it was Jo Grant who coined the phrase. (COMIC: The Heralds of Destruction) Another account suggested that the Fifth Doctor first heard it from River Song, who assured the Doctor that it would "grow on [him]" in time. (AUDIO: A Requiem for the Doctor)

When the Eleventh Doctor used it in front of the War Doctor, he ridiculed his successors, asking why they insisted on talking like children. The Tenth Doctor, embarrassed, claimed he "[had] no idea where he gets that from". (TV: The Day of the Doctor)

After hearing Rose Tyler say it, the Ninth Doctor was equally dismissive of the phrase. (COMIC: Doctormania)

The Tenth Doctor used it at least once after the Weeping Angels when trying to think. (PROSE: Prisoner of the Daleks)

The Doctor began using "wibbly wobbly, timey wimey" and variations with greater frequency following his regeneration into the Eleventh Doctor. (TV: The Eleventh Hour, TV: The Girl Who Waited, The Doctor, the Widow and the Wardrobe, The Angels Take Manhattan, The Day of the Doctor) The term lent itself to two other devices during this incarnation: the wibbly lever (TV: Time) and the timey-wimey distress beacon. (TV: The Wedding of River Song, Time) Amy Pond also used the term while the Doctor attempted to explain why the TARDIS matrix was unable to talk while inhabiting the TARDIS once more. (TV: The Doctor's Wife)

During the overnight invasion of trees, the Year 8 Gifted and Talented Group of Coal Hill School drafted a speech to tell the people of the Earth to leave the trees alone on the Twelfth Doctor's instructions. Amongst the chatter of the children, one of them said "timey wimey". (TV: In the Forest of the Night) Santa Claus mocked the Doctor for his simplification of scientific principles claming the Doctor would explain the situation as "dreamy weamy". (TV: Last Christmas).

Behind the scenes
In  released in 2016, Miles Morales references Doctor Who to Thor as an example in explaining how leaving the second Mjolnir will be found by future-Captain America and future-Thor and uses the phrase "timey-wimey stuff".