When did the transition from standard definition (4:3) to widescreen (16:9) for television anime happen?
According to user Zalis on the MyAnimeList forums, one of the first instances of widescreen anime was Betterman (1999). However, because it isn't one of the better known shows, it often gets overlooked and Vandread (2000) is often accredited with the first.

The cinema is often the first place for the widescreen transition of visual media, and early animated (and non-animated) films were sometimes shown in widescsreen at the cinema, and later reshaped for broadcast/dvd. The earliest example I could find for this was Magic Boy (1960~) This was  2.40:1 rather than 16:9 though. You can see the difference between the aspect ratios below:



I don't think there are any specific points in time that we can point to where there was a definite switch - just as the transition to digital happened, but the full transition seems to be almost done in 2006-2008, when 4:3 anime became minimal and even the larger series like One Piece, Naruto, etc switched over.

References: 


Anime &amp; Manga - Wikipress.
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