The Macross series is about The Power Of Idol Singer Music (and of reaction warheads). It should be noted that it usually has a different justification every time; only in Macross7 is singing an actual source of superdimensional energy.
The singer they got in for the English dub is said to have a similar effect on the audience.
In Macross Frontier, it's not so much the song as the passionate feelings behind it which affect the empathetic Vajra, who communicate through song (even across the galaxy, if the singer has the ability) themselves.
By the time of Macross Delta, they've explained everything in previous works as being due to Fold Waves, with the singers who were able to harness the Power of Rock in previous series having high Fold Receptor Factors which allowed them to generate said Waves while singing. The Idol Singergroup Walkure is specifically composed of girls who have this ability.
Mic Sounders the 13th of GaoGaiGar uses rock music from discs, microphones, and a key-tar from his flying soundstage to attack, disable enemies, and power up his allies. His Disc X is actually able to rip enemies apart at the molecular level or shoot a gigantic laser beam at them.
The Wave Motion Gun is powered by a heavy metal guitarist's "groove". However, it's also partial to evil techno.
It's worth noting that the original title is Kachou Oh-Ji ("Office Boss Ohji", referring to the main character). The subtitle? Hard Rock Save the Space.
The anime is all about this. An aged, retired rocker (and later the rest of his old heavy metal band) is called back into action by a group of aliens to fight off an evil alien space empire with, you guessed it, the power of rock.
The Science Ninja Team Gatchaman episode "Murder Music" had Galactor capturing a rock band, drugging and forcing them to play music personally created by Sosai X. With the recordings, a special ship with super speakers plays the music at a zillion decibels to create a devastating sound wave that can drive people insane and shatter matter.
In Ah! My Goddess, the demon Marller/Mara — on top of her weakness to good luck charms — can be forced to dance nonstop, and thus defeated, when she is exposed to Rock music.
Similarly, Belldandy's half-demon sister Urd, who is otherwise one of the most Bad Ass characters in anime, can be put to sleep by the Japanese music genre of enka. She hates this.
How monster users in Buster Keel! power up their monsters using guitars. Usually depends on what pick is used and what music is played.
The final battle in Shin Koihime†Musou is a war between Idol Singers. The Chou Sisters versus an impromptu group formed from Enjutsu, Choukun and Kakuka. The war is not so much about the singers as it is controlling their fanboys to either take revenge or calm down. The final blow is dealt by Ryuubi and company singing an a capella version of "Ima wo Shinjite" that wins by The Power of Friendship alone.
In Grenadier, one of the Jutensen uses a miniaturized organ as a weapon that can project an impenetrable shield of sound, and can probably melt your insides with a riff.
This plays a massive part in 20th Century Boys, so much so that one of the major plot points of the final arc is that of Kenji uniting the rundown populace of the now Crapsack World and giving them courage through his protest song that is being broadcasted via pirate radio. It also serves as a way to let his friends know he is still alive, due to modified lyrics in the song. In the end, Kenji and his song are one of the main reasons that Friend is eventually defeated.
One of the best examples of this trope in the series is when Kenji returns after 10 volumes and 15 in story years. He shows up at a maximum security checkpoint on the outskirts of Tokyo, and is promptly surrounded by armed guards with orders not to let anyone pass through. What does he do? He simply whips out his guitar and begins singing his song, which mystifies the guards. However, the peasants in the checkpoint recognize it from the broadcasts and begin converging towards him. The guards become agitated when they notice the effect the song is having on the peasants, so they quickly shoot Kenji in the chest. Five seconds later, he's back up on his feet and singing his song right from where he left off. Just at that point, hundreds of his followers emerge from behind him and easily overwhelm the guards. This leads Kenji to deliver one of the most memorable lines of the series to one of the guards.
In Fairy Tail, the Villain Vivaldus brainwashes Juvia with a magical electric guitar by playing various solos.
FLCL is set in a world where electric bass guitars are the most powerful objects in the universe.
Parodied in Bobobo-bo Bo-bobo. Denbo, Jelly Jiggler and Don Patch fight J by singing, which makes stuff in the song become real. But the songs they think up are completely ridiculous, involving things like Beta-Carotene and a 49 year old man. There's even a little DDR section. This is recycled in the manga sequel, except that Pokomi is the one singing.
Senki Zesshou Symphogear is about powering your armor through singing and kicking alien ass with it. This gets even better as the casts becomes larger each season, as each and every Symphogear is a capable singer of their own right, on top of being big names in the anime industry.
In Pokémon Special, Bianca plays a certain song on the guitar until Meleotta remembers Relic Song, allowing it to change Forme and fight off the flock of Vullaby attacking them.
The main premise of Show by Rock!! involves the main characters, who are in a band, harnessing the power of music to defeat monsters created by the Big Bad.