Soft locks...

Okay so, well duh, Pokémon Generation 1 is a broken, broken mess, so let's choose that.
Now, the easiest way to softlock someone is at the league because you need multiple HMs to get out of the area, now we need to look at the first E4 members team:
Lorelei has a Level 54 Dewgong with Growl, Aurora Beam, Take Down and Rest. That's perfect.
Now, next step, we need to have either a Fighting type or a Poison type, which both in the long run work to hinder progression but if we want a TRUE softlock, we need to choose either Mankey or Primeape to the point where they know the move Rage but still as low level as possible.
Now comes the annoying part of set-up, we need to deplete the PP of every move that isn't Rage (just fly to one of the first routes and punch some Pidgeys or something.)
Now that we have done that, we need to release *all Pokémon* that aren't Primeape.
Then we will remove all items that we can get rid of and spend all of our Pokedollars on Items we just resell again, until nothing remains. We box the key items that cannot be discarded.
Now we only have a Primeape or Mankey with Rage, it's time to move into the first room of the Elite Four, in which we save.
Now the lock is completed.
To explain why it works:
Gen 1 has multiple battle and move quirks that will now be abused:
First, the Trainer AI will always use the move whose typing is super effective against the typing of the Pokémon which you use.
This means Lorelei will only use Rest.
Secondly, the opponent AI's Pokémon will never run out of PP.
This means Rest can be used indefinitely.
Now, lastly, we have the move Rage.
Rage is a move that if it is used, will be used for an infinite amount of turns.
This results in an infinite battle as soon as Rage connects. The player can no longer do anything.

Now there is a way out, in theory, since every move had a 1/256 chance to miss in Gen 1.
Since Rage has 20 PP, the softlocked needs to miss 20 in a row, meaning the probablility is (1/256)^20. Should this somehow happen, they are still stuck at Indigo Plateau.

 

Hmm... I don't know if this counts, but there is a softlock for a very popular ROM hack series~

In Pokémon Adventure: Red Chapter, one should store all items and Pokémon except Poliwhirl/wrath who knows Surf, use said Surf on a small body of water (preferably one with low-lvl or no Pokémon) then use the Fashion Box. Congratulations, you are now stuck in the water~! The only way to get out is to make Poliwhirl/wrath faint, but depending on which water body you use, it will be hard to impossible since our Pokémon's Ability is Water Absorb~

Either that or just do PKMN Red/Green/Blue's Glitch City glitch and enter that 1-by-1 area~