Most Eurobeat songs. It's produced mostly in Italy and sold mostly in Japan, and of course, neither are English-speaking countries. English covers of Japanese songs, such as "Hot Limit", are especially gratuitous. A few singers, eg Domino, sometimes sing in Gratuitous Japanese.
That's because a lot of dance music (from non-English speaking European countries) have songs in English. To be fair, the English is often a bit better than what Japanese music has in regards to English song lyrics.
The French band Mademoiselle K has a song called "In English" parodying this trope. It's in perfect English but the lyrics go:
I wrote a song, My first song in English. I wrote a song, To talk about nothing.
Most Japanese Vocaloids are pretty bad at pronouncing any English word (except for Luka, who actually has an English voice bank). Try Miku's "Cinderella romance" (SUPAA POWAFURU SINDERRA, LOVU LOVU LOVU LOVU LOVULII DAHLEEN! SUPA POWA FURU LOVU ATTAKKU!)
The German a capella group Wise Guys sing a song remarking on this phenomenon in German, aptly titled "Denglish": "Oh Herr bitte gib mir meine Sprache zurück" slowly becomes "Oh Lord please give mir meine Language back" as the song goes on.
The theme song to the Anime version of Witchblade KICKS ASS! The couple of English lines are still gramatically terrible, though. ("It's all over" and "Break out" are OK, but "Give me your XTC" is just wrong.)
It's pronounced ecstasy, the song title, XTC, is just a pun.
Falco's new-wave/rap song "Der Kommissar" starts with him counting in English ("two three four") and then in German ("eins zwei drei"), and contains some gratuitous English in the verses. In the version that is loosely translated into English by After the Fire, they reversed the languages of the first part, so it starts "zwei drei vier" and then "one two three". The gratuitous English was left in English, but the refrain "Alles klar, Herr Kommissar" was kept, producing Gratuitous German.
Most of his (comparatively) better known song "Rock Me Amadeus" is in German, but also has bits of English (including the chorusnote He also calls "Amadeus" a "superstar").
Con te partirò. The lyrics are entirely in Italian, but in most performances a couple of lines are replaced with the English line: Time to Say Goodbye.
Japanese heavy-metal band Maximum The Hormone have a habit of invoking this trope regularly (as if their name wasn't evidence enough). Choice song titles include Policeman Fuck and Anal Whiskey Ponce, as well as lyrics regularly including English words and phrases that only just about make sense:
Saa tomerarenai eraser rain
Pink shambles speaker chu! mega lover, Aneki lover sign
A note about Japan's apparent love affair with the word "fuck". The combination of straights and curves it presents is aesthetically pleasing to them, so people who don't know what it means might throw it on somewhere just to have some nice-looking detail, oblivious to what American viewers are going to think.
The name of the Japanese rock band Bump Of Chicken is actually a translation error that the band decided to keep because they found it amusing.
Some even have entire songs in Gratuitous English. A particularly weird example is BeForU's "Red Rocked Rising English Edit.", which had a Japanese version first.
K Pop is a MAJOR offender of injecting meaningless English into its songs. It practically needs it's own page.
When she isn't singing in English, South Korean pop singer BoA's songs are still littered with English.
The band Exo's debut. "Careless, careless/Shoot anonymous, anonymous/Heartless, mindless/No one who care about meeee..."
SHINee deserve a special mention: "Fantastic, elastic, fantastic, elastic..."
Generally speaking, the level of Gratuitous English use varies from company to company. SM Entertainment (who owns both BoA and EXO) tends to use more than other music companies (i.e. YG Entertainment, JYP Entertainment, etc.), where the song writers actually speak a decent amount of English.
PSY breaks out into a chorus of "HEYYYYYYYYYYYYY, SEXY LADY" in his famous "Gangnam Style" video. There's also phrases like "baby, baby" and "You know what I'm saying?!" before the final chorus.
Rie Fu is known to subvert this trope, though living in North America for the first few years of her life did a lot to help her enunciation, the songs Life is Like a Boat, I so Wanted and I wanna go to a place are all good examples.
Joe Inoue is also known for this; in fact his Japanese is actually more accented than his English. The music video for Closer, the fourth Naruto Shippuden intro, had him having a fairly decent conversation, in English, with the resident leader of the town.
It should be noted that Joe Inoue was born Los Angeles, USA and is a native English speaker, something easily mistaken. He apparently claimed to have learned Japanese from watching Anime, arguably making him an Otaku and of course One of Us. This would also account for his Japanese being the more accented.
Anything by Laugh and Peace. Even worse is that the songs tend end up sounding incomprehensible instead due to the heavy accent.
They released "The Revenant Choir" as their debut single, which is written and sung more-or-less entirely in English, with only two lines in Japanese. Not that you could tell with Kamijo's heavy Engrish, which is only slightly better in the album's re-release, and the damn near incomprehensible lyrics ("It's a night when the moon laughs at lover", "poured crimson admiration into Holy Grail"). To make things weirder, the original release has an English voiceover at the start and end of the song by native speaker Leah Riegle.
Their third studio album, Holy Grail contains "Love will be born again", which is entirely in English. It's considerably more comprehensible than "The Revenant Choir", with the English and pronunciation being greatly improved, but there's still a smattering of Engrish in there. (However, more than a few fans have noted that Kamijo sings it better live, compared to when it was recorded for the album.)
Kaizers Orchestra mostly sings in Norwegian, but on "Die Polizei" they slip into gratious English.
The J-Rock band 403Forbiddena. Most of their songs are all in accented Gratuitous English, which makes it hard to figure out the lyrics for both English and Japanese-speakers.
Tommy Heavenly 6 has done this with at least one of her songs: "Black Paper Moon"
Fairy Blue kimi no tame ni
hoshi wo kudaki
Black Paper Moon!
The Japanese rock group Beat Crusaders sing exclusively in English even though none of them speak the language. This leads to lots of Engrish (a cover of "I can see crearly") and terrible grammar, but with song titles like "Joker in the Crotch", who can complain? Also, their songs are super catchy.
J. Quest (inspired by Jonny Quest) was supposed to be pronounced "Jay Quest". Eventually they spelled out the whole word for J in Portuguese, Jota Quest.
Los Shakers were a Uruguayan rock band of the sixties who were heavily influenced by The Beatles, and despite the fact that they primarily played to a South American audience, recorded most of their songs in grammatically shaky English. For instance, "Break It All" has the refrain "But when the music start / don't stand there like a fool / and break it all / you listen me, break it all".
Brazilian Black/Thrash Metal band Sarcófago are possibly the greatest example of this trope in history.
If you are a false don't entry
Because you'll be burned and died
The nuclear drums will chrush your brain
Slaughtering all with intensive pain
In Mexico, there was a very famous and respected musician and entertainer named Francisco Gabilondo Soler, who was best known by his character "Cri-Cri, El Grillito Cantor ("Cri-Cri, the Singing Cricket".note "grillito" literally means "little cricket".) This happens in his song "El Ratón Vaquero", when the titular "Cowboy Mouse" asks the singer to let him out of the "ratonera" (mousetrap):
"What the heck is this house / For a manly cowboy mouse? / Hello, you! Let me out / And don't catch me like a trout!"
Japanese band Alice Nine does this. It's very obvious in their song Blue Planet.
There exists a Cover Album that consists of classic punk songs being sung by famous Japanese voice actresses. Most of whom are not fluent in English. Draw your own conclusions as to how they turned out. Or better yet, let this set the tone.
Namie Amuro's "Neonlight Lipstick" is a weird case of this. The chorus and last verse are entirely in English, and the verses that do have Japanese involve a lot of code-switching between the two. It's not particularly bad English for J-Pop, but the song ends up having more English than it does Japanese.
A somewhat amusing case is "Elettrochoc" by Matia Bazar. Not enough that the only English word used in the song is "said", but the other singer also immediately translates into Italian ("dice"). Just in case an Italian hearer doesn't speak English...
i-dep's "Magic" featuring Cana (presumably on vocals) and "Rainbow" are entirely in English with dubious grammar (especially the latter). Between the accent and the autotune, you may wanna cue up a lyrics search.
Clazziquai Project has this all over the place with their lyrics. Sometimes it works well with the Korean lyrics, and other times it just seems like English for English's sake.
One or two of the songs by Kuba Sienkiewicz. "Leasing" is sung fully in English.