I haven't watched many that I would put in this category, and I have also not been into dramas for all that long, so take "classic" with a grain of salt here, but:

Rain or Shine is one that comes to mind. It's about two people falling in love while they heal from a (shared) traumatic event, and dealing with the after-effects of that event including their own survivor's guilt.

Chocolate is another that might fit the "melo" category - complex family dynamics, slow-burn romance, tragic side-stories (it's set largely in a hospice, so lots of folks are near their end), trauma to overcome, etc. Loved the main cast here.

And then, do folks count My Ajusshi as a melo? If so, it's my favorite (currently my third favorite drama I've seen). A young woman and a middle-aged man in bleak but wildly different circumstances collide. She learns how to be a person and exist in the world without constantly looking over her shoulder, he wakes up and tries to figure out what he values and what he wants his life to look like. It's totally beautiful and heart-wrenching and I want Lee Sun-kyun to read me a bedtime story every night. Some people found it a tough watch, but mostly it felt incredibly hopeful to me.

I think the defining feature of the classic k-drama melo used to be a sad ending à la Autum in My Heart or Stairway to Heaven. But I don't know if I'd recommend them because they're really not my favourite genre 😬.

- The ones with the four seasons : Autumn in my Heart and Winter Sonata (Scent of Summer and Spring Waltz are less famous I think)
- Lovers in Paris
- Memories of Bali

Secret aka Secret Love with Ji Sung and Hwang Jung Eum is the only one I've watched that I would call a classic melo. FL is in jail for causing a wreck that killed the ML's girlfriend.

I'm not watching Oasis so I don't know what it is about and what dramas it is similar to. But classic melos I would definitely recommend are:

Que Sera Sera
Scent of a Woman
Resurrection

You Are My Spring.
My Liberation Notes.
Find Me In Your Memory.
King of Baking, Kim Tae-gu.
The King in Love(The leads have a description quite similar to Oasis').
Temperature of Love.

I Have a Lover. Stay with it through the 1st few episodes, and it is worth it.

A thousand days promise
The Princess Man
That winter, wind blows
Missing you

I’m Sorry I Love You, and a couple of films: The Classic and Il Mare.

The classic melos I've watched have been completely addictive. But, maybe a more experienced watcher can define a classic melo---I'm not really sure. Can it be a historical since practically all of them could fit a melo theme. Does someone need to die for it to be a melo? There is something special about older k-dramas in the addictive/melo genre. I personally like going back to watch older shows as opposed to keeping up with the modern shows. I'm not sure what makes them so special, but I rarely hit the pause button on these classics. Something Happened in Bali, Stairway to Heaven, 49 Days, I'm Sorry I Love You, King of Baking. I recently watched Secret Love Affair which I think could qualify.

Giant is a great melo- but it’s 60 episodes long.

Damo- historical/melo mash

The Seasons drama: Winter Sonata, Autumn in My Heart, Summer Scent and Spring Waltz

Bright Inheritance featuring baby Lee Seung Gi and Han Hyo Joo?

😂 am I showing my age?!

Stairway to Heaven was CLASSIC MELO TORTURE...
: D

Ones I think haven’t already been mentioned:
-At a Distance, Spring Is Green
-Defendant
-Hello Monster
-It’s Okay Not to Be Okay
-Kill Me, Heal Me
-Padam Padam
-A Piece of Your Mind
-Just Between Lovers
-The Smile Has Left Your Eyes
-Greasy Melo (to me, more of a romcom with some melo elements)

Uncontrollably fond. Brace yourself for top contender for worst mother in K drama, and serious levels of noble idiocy. I went in blind because I did not know about the melodrama genre. Suffice to say it left quite the impression…

I'm Sorry I Love You
Ruler of Your Own World (not really sure if that's a melo, doesn't really feel melo, but I have to mention it)
Nice Guy (definitely melo, has the melo music)

This one's hard.
Star in My Heart
Glass Slippers
Angel Eyes
And although it's not a classic, not even sure if it's a Melo, but I just rewatched I Heard It Through the Grapevine.

Time Between Dog and Wolf
It’s an action melo. Has the best amnesia plot I’ve seen in a kdrama. Also features Lee Jun-ki & Jung Kyung-ho.

These two aren’t the classic 2000s melos but I still throughly enjoyed them:
Marriage Contract
I’ll Go To You When The Weather Is Nice

Youth of May!! I've watched it about half a dozen times and it stands as one of my favourite melodramas (though it feels a bit weird to call it a favourite considering the subject matter). It's really well directed, written and acted and it leaves such a deep imprint on you - I found myself thinking about it for a long, long, long time after my first watch.

Not for everyone of course but 100% worth watching if you're up for it.

Also another great one is Moment at Eighteen. The cast is phenomenal and I was so wholly absorbed and invested in the story from episode one. This one also really viscerally makes you feel all the feelings too - a sign of a good melo in my book - and has a beautiful cinematography to match.

Secret Garden
Kim Sam Soon
Scent of a Woman
Fated to Love you
It Happened in Bali
The Legendary Witch

Sandglass
Giant
Secret Love Affair

I'm Sorry I Love You - still powerful and unforgettable!

Here is some classic Jang Hyuk melo goodness:
Thank You
Robber
Fated To Love You
Money Flower

Maybe this is not a melo drama but I highly recommend “All about Eve”.

Melo is one of fav genre. But it doesn't mean I love all melo drama. My fav definitely Giant, que sera sera, the devil and the innocent man

I can't talk about very old dramas because the eldest I have watched are from 2009-2010.

But these are the k-dramas I have watched that could be melos, if I understand the genre:

- King of Baking. Melo or makjang or both? Lots of "unfortunate events" happen to the adorable Tak Gu (Yoon Shi Yoon).

- Just Between Lovers / Rain or Shine. Main leads having a difficult, sad existence and finding comfort on each other.

- Girls Generation 1979. I loved the classic style of this series. It combines sad and happy moments really well. Beautiful OST too.

- While you were sleeping. Not one of my favorite k-dramas but I think it's a fantasy-melo, isn't it?

- My unfamiliar family. Melo or slice of life? Anyway, I loved it. Really good script.

- 25-21. Beautiful series. Maybe the less "melo" among my picks.

Other suggestions: It's ok to not be ok, At a distance, spring is green (halfway between rom-com and melo), 12 Nights...

For classic melos nothing can beat Autumn in my Heart and What Happened in Bali. I got kdrama scars from these two, LOL. Very compelling, addicting and unforgettable. Followed by Winter Sonata, Lovers in Paris, Summer Scent, Lovers. No recent drama has surpassed these for impact, I think.

Glamorous Temptation (2015)
May Queen (2012)

"Winter Sonata" is a classic!

Agree with Snowflower that "I'm Sorry, I Love You" is memorable.

Save the Last Dance for Me. My first Ji Sung drama and it’s so underrated!

WINTER SONATA - started the K-tsunami!

The Princess' Man