While not a solo, The Hooters’ “And We Danced” starts out with a duet of two atypical rock instruments: a mandolin and a melodica. (The band’s name itself comes from a nickname for the melodica.)
Piccolo trumpet is used in several songs, including “Penny Lane” and in Barry Manilow’s “I Write the Songs”.
The Association’s recorder solo in Windy. Rock on, bro! 
The bassoon riff in the Harper’s Bazaar cover of the 59th Street Bridge Song.
And when you’re good, all you need is a cowbell.
Bzzzt. It sounds exactly like a trumpet, because it is.
Led Zeppelin’s Whole Lotta Love has a theremin bit in the free form middle section.
Minor Waltz, by the Asylum Street Spankers has a saw solo.
Great choices everyone. Thanks! I’ll take a listen to these as I get the time.
Also, hard to forget “New York’s All Right If You Like Saxophones”
(start at 2:19 unless you REALLY like Fear)
Played by Paul Tanner, I assume?
Wagakki band feature samisen, koto, wadaiko and shakuhachi solos live. Here’s a great bit of Beni Ninagawa on samisen.
While there are no solos per se, the shakuhachi, taiko and koto bits in songs like this do gives an idea of the quality of the solo performances in live songs.
Early jazz went in some odd directions sometimes.
In 1924 Clarence Williams’ Blue Five recorded Mandy, Make Up Your Mind with a solo by Sidney Bechet on an instrument most people have never heard of: a sarrusophone.
Runnin’ Ragged (Bamboozling the Bassoon) by Joe Venuti’s Blue Four, recorded in 1929, features a solo by Frank Trumbauer on (what else?) bassoon.
A Mug of Ale by Joe Venuti’s Blue Four, recorded in 1927, features Adrian Rollini on both bass saxophone and a sort of tiny clarinet he called the “hot fountain pen.”
Getting away from early jazz, Going Up the Country by Canned Heat features a series of flute solos.
Can’t mention mandolin without referencing Rod Stewart’s “Maggie May”.
The harpsichord occasionally makes an appearance in various songs; my personal fave is the uneven-metered (and oddly-tuned) solo in “Golden Brown” by The Stranglers. The whole song is pretty much unconventional.
The music for “Going Up the Country” was borrowed from “Bull Doze Blues” by Henry Thomas, who originally played those solo breaks on a type of panpipe called the quills.
Interesting song. Never heard it before, but it’s basically all in 3s, except for the harpsichord interlude, where it’s threes with a four every fourth measure (so 3+3+3+4).
I don’t think I’ve seen mention of the violin solo in CSN’s “Wasted on the Way.” Come for the vocal harmonies, stay for the fiddle! 
Saxophone solos were all over Motown songs. The Four Tops’ “It’s the Same Old Song” has a pretty good one. The sax solo that ends Marvin Gaye’s “Mercy Mercy Me”, played by Wild Bill Moore, is particularly beautiful.
Bingo. See here:
After listening to the recording, I’m inclined to believe that short “tiny clarinet” solo is soprano sax. My teacher used to refer to soprano sax as “weasel horn.” which describes it quite well. (No disrespect to all those soprano sax players out there, you know who you are.)
Lotsa of great mentions in this thread, I love that they’re all over the place musically. Dopers have big ears!
Oh, and the mellow instrument in Jackie DeShannon’s hit “What the world needs now” is euphonium, which is a valved tenor horn. Like a tuba but smaller.
Here's Maynard Ferguson taking a chorus on euphonium: (WARNING: Maynard was the "King of the High Cs" and he starts on trumpet up in the screaming range, so you might want to adjust your speakers/headphones so your head won't asplode. Just saying.) Yes, children. that's how we dressed in the Fabulous '70s. your humble TubaDiva[quote=“TubaDiva, post:77, topic:781626”]
Bingo. See here:
[/QUOTE]People have argued about the “Low Spark” solo for years. Yes, it’s sax on that live version, but on the studio version it’s Steve Winwood on organ played through a fuzzbox. (Note how it begins playing over Chris Wood’s brief saxophone break.)
Oh! French horn, Leon Russell, “A Song For You.” Sublime.
Looks like it’s both, doesn’t it? 