Estonian here.
* Worst song ever in the history of space (they were hoping to go to the Eurovision with this, and iirc almost succeeded; later they were actually invited to the Paris Fashion Week to perform this on stage):
https://www.youtube.com/watch?v=2dllo85ZSUk (Winny Puhh - Meiecundimees üks Korsakow läks eile Lätti / Korsakow, a guy from our area, went to Latvia yesterday)
* There is something very Estonian about this (not only the landscapes, it's actually a funny song, but very heart-warming):
https://www.youtube.com/watch?v=GVfPf006Vw0 (Orelipoiss - See alles jääb / Organ Boy - This Will Remain)
* A camp classic from the 1990s, sung and performed by a former chemistry teacher:
https://www.youtube.com/watch?v=SoZG75CTB7g (Onu Bella - Ma võtsin viina / Uncle Bella - I Drank Some Vodka)
* Something I discovered recently -- stunning conteporary folk. It's an old spell from Estonian folklore:
https://www.youtube.com/watch?v=7-Kpt0fFgME (Duo Ruut - Tuule sõnad / Words for Wind)
* Maybe the most popular contemporary folk group in Estonia these days:
https://www.youtube.com/watch?v=BP3wAfIIhAo (Trad.Attack! - Kallimale / To My Dearest)
* An influential acapella group:
https://www.youtube.com/watch?v=9cjuTYZBWOI (Estonian Voices - Kättemaks / Revenge)
* Another widely loved singer is Mari Kalkun, mostly interpreting older songs in the Võru dialect:
https://www.youtube.com/watch?v=AClRxNvbiZY (Mari Kalkun - Mõtsavele mäng / Forest Brother Game)
* Ah, and I'll also add Vennaskond, a punk (sort-of; hard to define actually) group that was strangely loved by everybody in the 1990s, inclunding those who generally hated rock music. This song is not typical for them, but I sometimes listen to this when I want to be sad:
https://www.youtube.com/watch?v=oppqjn1voWo (Vennaskond - Depressiiv-maniakaalne psühhoos / Manic Depressive Psychosis)