Classic proof of love, from one paramour to another: Singing a love song to them.
Comes in two varieties: The solo serenade, where it's just the two of them alone together, or the public affair, where a crowd of people is watching. The former is more intimate, the latter is more showy. Serenading someone in front of a crowd is a sign the performer isn't afraid to let everyone know they're in love. It's also a perfect set-up for a spectacular backfire if the suitor is unwelcome and/or utterly without musical talent. Solo serenades are often done under a balcony or outside window, usually with a guitar.
Any romantic song will do, but you get bonus points if the song has the girl's name in it.
A Sub-Trope of Grand Romantic Gesture.
A Sister Trope to Let's Duet, Old-Fashioned Rowboat Date.
Subverted and Played for Laughs in this◊ Quino strip. He imagines his loved one's reaction. Being splashed with a bucket of water but then being thrown a rose from her window. He sings for her, and gets the wet treatment and a cold shoulder.
Subverted in Strangers in Paradise, when Francine makes Katchoo sing "Love Me Tender" while Katchoo's locked outside their apartment.
Raina sings "Just Like Heaven" to Craig in Blankets.
10 Things I Hate About You has a great example of the public serenade, where Patrick "loans" school equipment to sing "Can't Take My Eyes Off of You" to Kat... in front of a stadium full of his peers. And then the marching band kicks in...
The above scene is parodied beautifully in NOT Another Teen Movie, where the lead character is told that to win a girl's heart, he should sing a song with her name in it (see page quote). He gets in front of the stadium full of his peers and sings "Janey's Got a Gun"; Janey is promptly tackled to the ground and tasered on suspicion of bringing a weapon to school. It doesn't help that he added her last name to the existing lyrics.
Although he doesn't sing per se, this is basically what Lloyd is doing standing outside Diane's window with a boombox in Say Anything.
Double Subversion from Get Over It: a drunk Burke sings "Allison" outside his ex-girlfriend Allison's window, but he does it horribly and she is disgusted. Later in the film, Allison's new boyfriend and Burke's rival, Striker, sings a much better version in front of a crowd of people.
The Steve Martin remake of Sgt. Bilko, has the army glee club sing "Only You" to his love interest.
There is the famous "You've Lost That Lovin' Feeling" scene from Top Gun.
In Meet the Feebles, Robert the hedgehog serenades his love, Lucille, by playing a song with her name on a mandolin. It is easily one of the least disturbing scenes in the movie.
A very young Elijah Wood sings "You Are My Sunshine" to a girl that he has a crush on in Forever Young.
Jerry does this three times in Top Hat: Sandman Dance, Isn't It A Lovely Day and Cheek To Cheek.
Three different versions of Orchids In Moonlight is used in Flying Down To Rio. The first is a solo piano piece (no lyrics) from Roger to Belinha as he's making it up (which leads to their first kiss). The second is when he's made it a huge musical number for her. The third time is when Julio, who also loves Belinha, sings her the lyrics and tells her how the song could make any woman fall in love with any man (referencing how she's fallen for Roger), but his version does receive a hug and her begging him to marry her.
In The Nutty Professor 2: The Klumps, Sherman tries to propose to his colleague with the help of a Mariachi band who repeat (in song) everything he says. It's very sweet...until Buddy Love takes over and starts making VERY awkward double entendres, like "Why don't you come over here and let me put the beef in your taco?"
Happy Endings, wanna-be cool guy Dave has written at least two songs for his fiance/then girlfriend Alex. One called Love to the Power of Love, and then this one he plays in Fowl Play Date.
Dave Al I miss you, you're the only one
I wish I knew how to make you come
home to me, home to me
come, come, come!
As a music-based show, Glee has this happen a few times.
An awesome instance happens when Puck sings "Fat Bottom Girls" to a plus-sized girl he's trying to impress. Even the resulting Epic Fail is awesome.
The Valentine's Day Episodes has Kurt's crush Blaine admit he wants to sing a love song to a "special someone" Kurt believes is him. It turns out a) Kurt is not who Blaine wants to sing to, b) Kurt has to help Blaine sing his love song to someone else, and c) the "love song" is a particularly racy Intercourse with You song, and d) the serenading ends up getting Blaine's crush fired from his job. Serenading your lover in Glee is Epic Fail indeed.
Blaine, backed by the Warblers, appear at WMHS to sing "Somewhere Only We Know" to Kurt after he moves back to Mc Kinley!
The episode "Rumours" has Santana singing "Songbird" to Brittany.
Santana: I've been going through that Rumors album and I found the best song that really goes one step past Landslide in expressing my feelings for you. Private feelings.
Santana: He's just furniture. Sorry, no offense. (Brad shrugs)
In the prom episode, when Artie serenades Brittany with "Isn't She Lovely?" as an apology after calling her stupid and a kind of "promposal." She accepts the apology, but not the invitation.
Mercedes: I thought this song was about a baby?
In one episode of La Catrina, Carlos serenades Jamie.
In the Black Books episode "Fever", Bernard attempts to woo his 'summer girl' Alice with a serenade. On accordion. Alice isn't interested to begin with...
Bernard: Hello! I've come to serenade you. I can't play guitar. I can't play this either, but I thought it'd be less obvious.
James May's Man Lab dedicates a segment to learning the art of the serenade; James teaches researcher Charlie how play the songs of Thomas Campion (1567-1620), and a way to cheat at playing the guitar. Charlie later woos fellow researcher Cassandra and asks her out to lunch.
Buffy the Vampire Slayer. "Under Your Spell", Tara's song to her fellow witch and lesbian lover Willow in the Musical Episode "Once More With Feeling". Subverted when Tara sings a Dark Reprise after discovering she literally is under Willow's spell; having had her memory altered by a forgetting spell. The two break up in the following episode.
Played With in the pilot film of a British series Cold Feet when Rachel essentially invokes this trope. Her long time boyfriend breaks up with her because of a job out of town. She's broken-hearted but very soon meets Adam who is absolutely taken with her and they start dating. He's madly in love with her and asks what he can do to prove his love. Rachel says she wants him to sing her a serenade. While he's naked. With a rose in his butt. They laugh it off. Some time later, however, her ex returns and begs her to take him back, which she does, breaking Adam's heart. Cue Adam, singing a serenade for Rachel under her window, completely naked and with a rose in his butt. Adam wins and she dumps the jerk for good.
Ally McBeal: Characters were fond of going to a bar where they could perform. Sometimes couples performed together or a song was a birthday present from a friend but Serenading One's Lover happened from time to time.
John sang a rock song to Kimmy who he dated briefly. He told an unfortunate half-truth, half-lie that he used to sing at school. He was very anxious, very adorkable, and not actually a good singer, but the ladies from his office asked everybody to cheer him up and applaud him, and it helped to create a believable aura of a rocker. Kimmy was thrilled and loved every bit of his performance.
Ally's boyfriend Larry sang her "Every Breath You Take" for her birthday. With Sting himself. Even Birthday Hater like Ally, depressed even more that year because Larry seemed to forget about it, defrosted and loved it.
The Big Bang Theory: Howard sings a crazy love song for Bernadette when they are broken-up, accompanying himself on the keyboard, in Cheesecake Factory where she works. People seem a bit embarrassed but it turns out she loves it and they get back together.
How I Met Your Mother: Played for Laughs and Played for Drama. In a flashback, hippie!Barney sings and plays a song for his then-ex-girlfriend Sharon who broke his heart, leaving him for a wealthy suit. He actually videotaped it, and he was half-singing, half-crying, begging her not to leave him. His friends see the tape and mock him, being used to his usual cool and aloof self. But they feel very bad when they find out how badly Sharon screwed him and that he considers this the most humiliating thing that has ever happened to him.
Krazy Kat loved this trope, and often used a "mandoleen" and a balcony as shorthand for romance in general.
Theatre and Opera
The Barber of Seville: Act I - Almaviva hires a band of musicians to serenade Rosina below her balcony ("Ecco Ridente in Cielo"). A bit later, he tries again with Figaro's assistance ("Se il mio Nome") to more success.
Some stagings of The Merchant of Venice have made Lorenzo serenade Jessica during the "elopement" scene. Back when adding stuff to Shakespeare when you edited for direction was considered perfectly fine, one copy of the script had an entirely original song for Lorenzo. Another simply stole a few lines from A Midsummer Night's Dream and had him sing them to her.
"Have you noticed how popular synthesizers are getting? It's not the same as a lonely man with a pony tail, singing of his love on a guitar. At the door of a fair maid's hacienda. But it is not far short."
When Principal Skinner was trying to win back Edna, he had the school glee club sing "Oh Edna K" to the tune of "The Lion Sleeps Tonight"
In another episode, Homer gets "Weird Al" Yankovic to serenade Marge with a parody of John Melloncamp's "Jack and Diane".
Subverted when Kirk Van Houten serenades his ex-wife with "Can I Borrow A Feeling". She's completely unimpressed and the divorce stands.
Parodied on South Park: Stan is trying to get Wendy back, and one of Wendy's friends suggest he stand outside her window and play a Peter Gabriel song as a romantic gesture. He chooses "Shock the Monkey".
In the cartoon "Solid Seranade", Tom woos his sweetheart by singing "Is You Is Or Is You Ain't My Baby" on the bass. Jerry and the local guard dog are not amused.
In "Texas Tom", cowboy Tom plays the guitar and sings to a very pretty kitty cat.
In the first episode of the FOX run of The Critic, Jay tries to convince Alice not to get back together with her ex-husband Cyrus, who happens to be an expert at wooing her back with serenades. In the end, Jay attempts to counter-serenade him on accordion, which he admits may have crossed the line from cute to idiotic.
In the Tex Avery CartoonLittle 'Tinker, B.O. Skunk, desperately in search of a lover, follows the suggestion of a book that cupid himself gave him and tries to serenade a female raccoon at her balcony. It almost works until she smells him.
Go on YouTube. Find any song that features a girl's name prominently. At least one of the comments is bound to say either "I played/sang this for my girlfriend, her name is X", or "my boyfriend played/sang this for me because my name is X".