I can't help you with popular music, but can comment on some old concert music.

Recorded music has changed things forever. We can now bring back performances from the past, we can almost travel in time and listen to what people were listening to in the 40s or whatever. We have photography and video... As /u/Keltik said, nostalgia...

Did popular composers ever bring back the romantic classical era, or did Bach-style fugues wax and wane in popularity over the decades?

The great J. S. Bach was quite fond of the olden ways. His work contains some amazing counterpoint, that is a way to compose music that has existed since the so called middle ages. Polyphony has a long and glorious history... But that was not what kids were doing in Bach's time. While J.S. was taking polyphony to great, dense, super intellectual levels, people had been getting into homophony for a while. Did he wanted to bring back a trend? Not exactly, but he was into old stuff.

Mozart became serious about studying the work of counterpoint masters. The education of classical musicians included counterpoint (perhaps not in the same way Bach's did), but Mozart and Beethoven got into it much more seriously in their later days. The result? Listen to Mozart's Requiem, and now listen to this one by Jan Dismas Zelenka (a great Czech baroque composer). Zelenka died about 10 years before Mozart was born.

Did Zelenka have a time machine to visit the future and get ideas for the Introit? I think it was Mozart who went old school (but not full baroque). There are obvious differences in his earlier music and what we see in his later works (a terrible, terrible thing he died in his 30s. He was just starting to write good music!)

The period we call classical in music happened while every other art was into Neoclassicism. Their neo is a reference to Greek and Roman art. We don't really have that in music... So, the other arts were into oldies, bringing back the good stuff.

In the origins of opera we see some well read people (in the 16th century) trying to bring back the drama of classical times (that is, Ancient Greece and Rome). They invented a new for of art by looking into oldies (of which they didn't know terribly much; we currently don't have a super clear idea what the Greeks were up to, either). Here you can listen to some Monteverdi, that would be Nero (as in, the Roman Emperor) and Poppaea Sabina singing.

During the 19th century people REALLY got into reviving old music. It was fashionable to present old music by unknown composers in recitals (the more obscure, the better). People were playing new music or something from the canon, but they included the music of obscure composers, such as this unknown J.S. Bach dude. Felix Mendelssohn campaigned very hard to bring back to the spotlight the great works of Bach. Hard to imagine, but Bach was not a god back then. His music didn't completely die with him, but he was not a household name until after the efforts of Mendelssohn and others. So, make sure to make a toss for good old Felix the next time you go out (and listen to his music, the bloody Nazis made a campaign to make him and others less popular, but Mendelssohn's music is fantastic).

Bach is perhaps the most famous case, but many composers have been brought back from the grave, so to speak. The romantics were very fond of that. They brought back this music with their 19th century instruments and musical practice, so they were kind of making covers of the old masters. Almost two centuries later we think we have something closer to what Bach was doing.

This is a pastiche (a work that imitates a style) by Reynaldo Hahn (1874-1947). He was a Venezuelan composer living in Paris, some of his music includes elements from the baroque and classical. The lyrics from that wonderful music comes from a poem by a great 17th century French poet (search for translations if you don't speak French, this is panty-dropping stuff), and he includes some baroque things here and there (but it's not 100% baroque). You will probably recognize this influence. Listen to this sonatina, also by Hahn. It includes a lot of late baroque and classical things (from the time of Bach and his sons, to that of Haydn). Yeah, it's not 100% classical, but there are elements of it.

If you want full on bringing back retro music, there's what we call Neoclassicism in music (not to be confused with everybody else's, for us it's NOT about the Greeks and Romans). Igor Stravinsky was a super revolutionary Russian composer, he did all kinds of crazy things. It was the roaring 20s, and this was their new music. What could Stravinsky do to make people turn their heads when everybody was already used to the craziest and most radical ideas? Well, a ballet. A ballet full of... very conservative old music! What better thing for the first section of a ballet in the 1920s than music by an obscure composer from the middle of the 18th century. Verbatim! Good artists copy, great artists steal, said Picasso (who lived at around the same years as Stravinsky, and is comparable to Igor in influence). You can listen to more of the music Stravinsky used for Pulcinella in this video.

Wendy Carlos took the music of Purcell and Bach, and "switched it on." She used synths to transform this old march (Purcell, 1695) into this magnificent electronic title music. She also worked with Bach's music, making it sound like it had never sounded before (I think that's not hers, but it's close in spirit, and it rocks). Fancy some Beethoven? Wendy reworked some Beethoven for Kubrick's A Clockwork Orange. Kids these days and their electronic music, we (in the world of classical music) liked it way before it was cool...

Michael Nyman is a, living, famous English composer (and musicologist). He was the first (or at least among the first people) to use the word "minimalism" (in the late sixties). He studied quite a lot of 17th century music. And it shows! I mean, he is REALLY into chord progressions reminiscent of that period. Here's an example. Is this 17th century music? No, but some influence is there.

We now have huge amounts of recorded music (and images, and video, and a hell of a lot of other resources), available to everybody thanks to computers. Radio and recordings changed music in many ways, and we are still experiencing changes.

We have few examples of musical notation before the 9th century, and early notations close this century are kind of problematic. We start to have a more detailed knowledge of music as we get close to the 13th century. Most of the periodization (the attempt to divide time into named blocks) and canon we currently talk about started around the 19th century (that when people said "this is baroque, this is classical, and so on; the renaissance tag was assigned earler). Music did not always have a canon, or a commonly known set of periods, that comes from the 19th century.

Musicians were not terribly into oldies. For most of history people were interested in new music, and it was problematic to have detailed knowledge of music from the past. How do you keep music alive without notation or recorded performances? Oral traditions, of course. But then you are part of that music and it's kind of hard to distance yourself from it... How much music can people remember? How much music can they be exposed to without recordings or other form to store it?

Music history became much stronger than before during the 18th and 19th centuries, and what was studied was limited (popular music, or the practices of other cultures were not on the same standing as "proper" music. The scholar study of popular music is still some times seen as uncommon, compared to the huge amounts of work done on the music of "dead white men"). The works of previous great masters were of course available in some form back then, but music from the past was not as widely studied as it is now.

Today, a kid with a tablet has access to more music than what conservatory students had available 30 years ago. That kid also has access to a free library of sheet music beyond the wildest dreams of the great old famous composers, or even the musicologists from just some decades ago. We cannot compare these recent "cycles" with the revivals and influences found previous centuries.

What we now know as nostalgia was not really possible until the advent of mass media (radio, film, recordings) after WWI. Before that it was just memories.

Nostalgia used to work in 30 year cycles. People in the '20s were nostalgic for the "Gay '90s" (you can see the tail end of this phenomenon in Mae West films). There was considerable nostalgia for the "carefree" (people always think the past was carefree), predepression, prewar '20s in the '50s. Part of this was in reaction to the new sound of rock & roll.

The '60s, incredibly, actually saw nostalgia for the depression '30s. This is when the blues and folk revivals renewed interest in artists like Robert Johnson and Woody Guthrie. Also by this time movie studios had sold their film libraries to television, giving second life to long-forgotten cultural icons. One example: the enormous increase in interest in the classic movie monsters (Dracula, Frankenstein, Wolf Man, etc) after Universal sold its old films to local TV stations (many of whom would build "Chiller Theater"-type series around the packages). Later in the decade Bonnie and Clyde made the era seem like just one madcap bankrobbing romp.

The '70s is when the cycle sped up. While there was some interest in the '40s (the Harry Truman fad, the Andrews Sisters in a hit Broadway show, the first season of Wonder Woman which was actually set in the '40s) this was the Happy Days/Grease/American Graffiti era. Indeed you could actually call the '70s the Nostalgia Decade. Not until the latter '70s would the decade produce a real culture of its own, and disco is hardly something to be proud of.

Although the '70s were the decade most dominated by nostalgia, the following one may have been the most fascinating, since the '80s actually saw nostalgia for the '60s -- yes, the tumultuous, divisive, violent, Vietnam '60s. I remember reading an interview with some teenager at the time who said she wished she could have lived then since "things were much simpler and people didn't have to worry so much" (!!!).

There is a marketing principle that says a new mass audience comes into existence every 7 years. And since so much of popular culture is geared toward that youth demo, this speeds up the cycle more than ever. Are people nostalgic for the "old" Spiderman of Tobey?

Also, today people have their favorite movies and music just a click away, so it never really leaves them. It's hard to be "nostalgic" (in the 1950s sense) for something you're never really without.