Here's an eclectic selection that probably reflects what I've been listening to recently as much as anything else. I'd add links like I did for your last question but my answer got shadow banned by yahoo so I'll leave them out and see if this gets posted:
Literally everything by World Party except for "She's the One" which was stolen from Karl Wallanger while he was in hospital after having a stroke, and it's nowhere near one of his best songs. (People say he should be grateful that Robbie Williams made it famous but as he pointed out nobody came to see him in the hospital and that's a terrible way to find out who your friends are.) Karl's songs are very poppy on the surface but the more you listen the more there is going on musically.
Gene Loves Jezebel "Heavenly Body" Fans say that this band should've been bigger but the truth is that they (both versions of the band after the twins split) are very uneven in their output and waste a lot of energy fighting, but when they're good they're very good indeed.
Simple Minds "this fear of gods," okay, the whole of Empires and Dance. Before they were bloated, old gits they were kraut rockers. If you only know their stadium music days you'll be shocked. This is proper sweaty nightclub music.
If you've not come across Kate Tempest she's worth the listen in a darkened room with good headphones on. She lives in that space between performance poetry and rap and is utterly mesmerising.
Gavin Friday "Caruso" He's been weird and wonderful since the days of the Virgin Prunes and what do people know him for? "Angels" and they think it's bloody Robbie Williams <sigh>
This song doesn't get played on the radio, it's literally the chorus! Amanda Palmer "Map of Tasmania."
The Bolshoi "Away"
John Cale "Catastrofuk" John Cale is a living legend People seem surprised that he's still alive and making music. I quite like his "old man music." He just doesn't give a **** what anyone thinks anymore and it's wonderful.
Kent "Pärlor" They were huge in Scandinavia. Probably the best band the Nordic countries have ever produced. I saw them in the early 90s at a student union and knew that if they could get a toehold in the industry they were going to be huge - and they were. I don't think they're that well known outside Scandinavia. They did release English versions of their albums, but writing in translation never sounded as comfortable to them and I think they thought it was a bit of a bother. They used to rewrite their lyrics in English to make it fit the music, but sometimes this resulted in the song telling a different story, so you'd get weird youtube comments on the Swedish language version of a song that make no sense at all unless you've heard the English. The music still rocks though.
The Alarm "Where Were You Hiding When The Storm Broke" Now here's a band that definitely should've been bigger than they were. They're still around, sort of, but they're more like a tribute band to themselves these days.