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For astute fans, Mr. Harman actually sings more songs than just the one that is most famous. I'm going to try to track them down, I know there are at least THREE.
Anyone who wants to contribute is more than welcome.
I will start with Mr. Harman, but it would be great to list any and all songs sung in the series - For example, Mr. Grainger, Mr. Lucas, AND Mr. Humphries all take a moment and sing "Mr. Umbrella Man" in S3E8. In fact, I'll make that #1:
#1 - "The Umbrella Man" sung by the entire men's staff in S3E8 while they are trying to come up with new sales ideas for the store.
#2 - "Rollin' round the world, looking for the sunshine" - Sung very often by Mr. Harman when he comes onto the floor. One example is in S4E5 "Fifty Years On" when he brings out the display with the hat that raises up and down. The lyrics I found online actually differ slightly from what Mr. Harman sings.
In one episode he finally makes it through the whole first line, saying: "Rollin' round the world, looking for the sunshine, everybody comes your way", which is different from the two songs I found when I looked it up recently.
I like Mr. Harman's version much better.
#3 - In S6E1, Mr. Harman is busy trying to finish up the manikins while the sales staff comes back from their holiday.
As Mr. Harman leaves the floor, he sings:
"Hold my hand, I'm a stranger in paradise"
#4 - In S7E2, Mr. Harman brings a chair onto the floor for Mr. Grace to sit in. While walking onto the floor, he sings:
"Isn't it nice to have friends to rely upon, you must have friends"
#5 - In S6E5, Mr. Harman is busy vacuuming at the beginning of the episode. Mr. Rumbold asks what he's doing, and Mr. Harman hilariously explains the whole concept of vacuuming to Mr. Rumbold.
Before Mr. Rumbold comes in, Mr. Harman is singing something like:
"Oh foggy day....in London Town"
Funny side note: I did not realize that "Goblin" was really a name for a vacuum cleaner, I thought it was a gag for the show.
Later in Grace and Favour (S1E5), there's a joke about Mrs. Slocombe using a "Junior Goblin" to vacuum the stairs when the staff is trying to prepare for their first guests at the hotel.
It was then I finally looked it up and found out it was a REAL vacuum cleaner company.
#6 - At the beginning of S6E1, Mr. Harman is singing something like "You need a hand, to hold someone would love you".
This seems to be a different song than post #3.
#7 - During S3E6 "German Week", Mr. Lucas and Mr. Humphries sing a song just before they bang their beer mugs together.
They sing the line: "Drink drink drink, two eyes that are bright as stars".
It comes from the song "Drink Drink Drink" by Mario Lanza.
As many have pointed out, it's also funny that the gag with the mugs did not seem to happen the way they rehearsed.
#8 - In S7E3 "The Apartment", Mrs. Slocombe is filling up a "heating pad" for her back with hot water from the sink while getting ready for bed.
While she fills it up, she sings something like "ease my heart" while humming a song.