Many bands, never had the recognition they actually deserve, and those bands below are definitely among them.
(In alphabetical order)
1) The Alarm
Welsh Alternative Rock band which had some moderate success during 80s, with songs like Rain In The Summertime, 68 Guns or Sold Me Down The River. Sold Me Down The River managed to reach #2 in US Mainstream Rock charts, and Rain In The Summertime #6. Yet they never made it in the Top 10 in UK, although they had 8 Top 40 singles. Even though, they never became as big as other Alternative Rock bands of that time like U2 or even Simple Minds, and since their Raw album in 1991, their popularity started to decline. So I consider them underrated.
2) Big Country
A very similar band, with The Alarm I mentioned above, since both had Punk roots, and they often included Folk elements in their sound. Slightly more successful than the first though, since they managed to reach Platinum status (in the UK at least. In US it reached Gold.), with their debut album The Crossing (including their US Top 20 hit, In A Big Country), and a series of Top 10 UK albums, they failed to make a big impact overseas. Even though they went to be highly successful in Ireland and in the UK during 80s, unfortunately in most countries they fall under One-hit wonder category, with their song In A Big Country, or the song Look Away in some others.
3) The Church
Another Alternative Rock band, from the same era, but this time they hail from Australia. They became known worldwide with their song Under The Milky Way, from their album Starfish from 1988. Apart from this song, they had a few other minor hits in Australia, yet in US they reached #1 in 1990 in the US Alternative Rock radio stations, with their song Metropolis. Although they are relatively known in USA and in Australia and New Zealand, and surprisingly enough their first 3 albums charted in Sweden, they failed to make an impact in most parts of the world, apart from their hit Under The Milky Way.
4) Dr. Feelgood
An English Pub Rock band, and one of the most known ones among this underrated genre in Rock music. Pub Rock was one of the main influences of Punk Rock genre as well. Anyway about this band now. Although they never had any major success (Milk & Alcohol, was their most successful single, reaching #9 in UK Singles Charts), and they never made a name outside of the UK they can be considered a cult band more or less. Apart from Pub Rock, they shifted into an AOR direction during mid-late 80s, mostly in their album Classic, which was equally good with their 70s Pub Rock albums.
There are not many bands from New Zealand, but Dragon is one of them. And like most New Zealand bands, they failed to make a worldwide impact, and remain popular only in Australia and New Zealand area. Rain was their only single that made it in US charts, yet peaking at only #88. They reached #1 spot in Australia though, with their song Are You Old Enough in 1978, and #2 twice, with their songs April Sun In Cuba and Rain in 1977 and 1983 respectively. A nice band, which never had the success they truly deserve outside the Australia/New Zealand area.
6) The Fixx
I can't tell if it's proper to include them, since they started as a New Wave (or even Synth-Pop as well) band, yet since their album Walkabout they shifted towards a Rockier direction and the Rock element was more dominant since then. But they are underrated any way. Not so much in the USA and Canada where they did well during the 80s, but anywhere else. Since their 1991 album Ink their popularity started to fade away in the USA and Canada too, and they disappeared from the spotlight since then, even though they are still active.
7) Golden Earring
Actually most people aren't aware by the fact, Golden Earring is perhaps the band which have outlived any other Rock band, including Rolling Stones. They exist since 1961, without going into hiatus since then. But unfortunately even though they're very popular in Netherlands (their homeland), they weren't so much successful anywhere else. Still, they managed to make it in the US charts with 2 hit singles. Radar Love which reached #13 and Twilight Zone, which reached #10. The latter was a #1 hit in the US Mainstream Rock charts as well.
8. The Hooters
Pretty unappreciated band, especially in their homeland (USA). They were very talented since they could play many different instruments (and include them in their recordings as well) except from the standard instruments in Rock music. Their first album, Nervous Night did well in the USA though, featuring 2 Top 10 Hits in the US Mainstream Rock charts, particularly Day By Day and And We Danced. All You Zombies almost made it in Mainstream Rock charts as well, peaking at #11. Since their second album, One Way Home, their popularity in their homeland started to decline, but they started to gain more popularity in Europe. Still they are considered a cult band in Europe, and not superstars.
Well like The Fixx I mentioned above, they started as a New Wave/Synth-Pop band (and their most successful period as well), since their album Man Of Colors they shifted towards a rockier direction (like so many other bands, that started as New Wave or Synth-Pop). And like most Australian bands, they failed to make an impact outside of Australian/New Zealand area. Only their albums Primitive Man and Man Of Colors, managed to make an impact in UK/Europe and USA respectively.
10) Inspiral Carpets
Sure, they were very successful during the early 90s in the UK, but they were unknown outside of their homeland, apart from a Top 10 US hit in Mainstream Rock charts, particularly Two Worlds Collide. Unlike most British bands of the 90s, Inspiral Carpets stayed away from the Britpop mania that was dominant in the UK during those years, featuring some kind of 60s Rock sound with 90s production techniques. Sure they deserved more recognition than they were given.
Whilst Rock music started to go downhill since the 90s, Levellers did very well. A band which sounded more like bands The Alarm or Big Country I mentioned above, instead of the overall Alternative Rock sound of that era.They were fairly popular in the UK area during the 90s and in some other European countries as well, yet they failed to make an impact overseas. Also, besides they didn't have any Top 10 hit single, their albums Levellers, Zeitgeist and Mouth To Mouth reached Top 5 in the UK album charts, and Zeitgeist reached #1 particularly.
Very well known in Canada and in the USA during the 80s but surprisingly enough they had zero success in the UK and in Europe, apart from their album Keep it Up which reached #25 in German charts. Apart from it, they had no successful singles, in the European continent. It's a shame though since I believe this band could be as big as Survivor or Journey. Perhaps their song Turn Me Loose, which was their biggest hit, might be known in Europe even if it didn't chart. But I'm still unsure about it.
13) The Silencers
A Scottish Alternative Rock band, which sounded pretty much like The Waterboys (I will mention below), particularly in their early albums. Although they never managed to score any significant hit single and none of their albums charted highly, they still have a decent amount of fans. Painted Moon was their biggest hit, yet it peaked only at #57 in the UK charts, but managed to make it in the US charts as well, although at a low spot again (only #82). After some point during the 90s they included more Electronic elements in their music, which they didn't suit them in my opinion.
A German Progressive Rock band, which had moderate success during the 70s, but only moderate. Spartacus was their most successful album peaking #27 in the US album charts. Although it's considered a decent spot for a non-American band, this album is considered a lost Progressive Rock classic which failed to achieve an all-time classic status. Well, since they were more of an albums than a singles band, I will post their Spartacus album whole.
15) The Waterboys
Besides the act they have a strong fan base, they never became Stadium Rock superstars even if they were very talented and most musicians who were part of this collective (they are more a collective than a band I believe, since most members weren't stable except frontman Mike Scott). Still they had the potential to be, judging by their first three albums which are heavily underrated. Their single The Whole Of The Moon, had a moderate success in the UK and in Australia and New Zealand though (the 1991 reissue reached #3 and #2 in the UK and Ireland respectively). They started to gain more popularity with their album Fisherman's Blues where they followed a more Celtic/Folk Rock direction and the title track was a hit in the US Modern Rock charts peaking at #3. Also their 1993 single The Return Of Pan made it in the US Modern Rock charts at a decent spot (#10).