Here's a thing I've noticed: new technologies sometimes mimic the kind of "mistakes" that older technologies allowed. It as if, having created and mastered a new medium, we long for the old days and the errors our old gear used to make, so we copy the most vivid of them into the new.
Note what happens about 54 seconds into this lovely movie of the artist Chris Piascik -- what is that? It's film trailer stock, added by the filmmaker, perhaps to underscore the sense of craft revealed in his behind-the-scenes look at an artist at work. But, it's a little goofy, because not only does the director show it "accidentally on purpose," but he's not working in film at all: it's a video. In fact, a lot of this video has washes of discolor that look something like film that's been overexposed or had a bad wash in the chemical developer, but no film was harmed in the making of this short movie.
So this is a vestigial inclusion, a knowing nod backward as the art form of movies jumps from one media, film, to another, video. Is it because the old media had more characer, even if it was more complex and less forgiving?
Photographers seem to love this. The look of Polaroids is popular: check out the Fake Polaroids group in Flickr; some digital photo albums will place your images in a Polaroid frame, or even break it up like a David Hockney photocollage. Another group on Flickr exposes the 35mm sprockets, and this Lomo photographer created fake vignetting.
TiltShiftMaker can turn an ordinary photo into one that looks as if you took it with a special lens designed for another purpose entirely (which lens itself mimics an effect possible with older view-type cameras, where you could shift the film frame).
Away from video and photo works, other examples exist. For instance, it's common to hear the sudden scccrrrrttcch of a needle on a record being inelegantly grabbed when some action has to come to a sudden, unexpected stop, like when Dad storms into your party and tells all your friends to go home. For instance, near the start of the movie A Series of Unfortunate Events, the fake squeaky-clean intro scratches to a stop (ffwd to about 1:44) to reveal the movie's real (and much darker) beginning. This sound has become an accepted trope even as vinyl disappears. (Well, of course, vinyl is still in use by hip-hop turntablists, obscure sample-seeking DJs, and Neil Young, but you get my meaning. How many 12-year-olds have ever used a turntable?)
Another cliche occurs whenever anyone in a movie or TV show has to make a speech into a microphone: the short, uncomfortable burst of feedback. This vestige is more about our former ignorance of technology, but after fifty or so years most public-address systems work pretty flawlessly. Here's a scene from Romy and Michele's High School Reunion (ffwd to 0:34). This wasn't even used to create a moment of embarassment or to attract the crowd's attention. It just occurs to let us know there's a PA system.
Others? What about an iPhone app that lets you use a rotary dial to make phone calls? (It's not even well done: higher numbers like 8, 9, and 0 should take longer to dial.) Or adding sounds to otherwise silent electric cars?
I'm sure there's more of this out there, now that I have my eyes open a little.