The problem with singed vs [insert melee champ] (P.S. he's not even close to BROKEN people)
The problem with singed is that if he has half a brain he is legitimately unpunishable. If you ever try to trade or all in a singed he inevitably just flips you and runs away.
Of course, this insane safety also translates to you being able to generally ignore singed in lane and play purely defensively which makes it hard for him to kill you, but this ends up with the lane being awful to play as it's essentially two people farming knowing that they can't do shit to each other. Singed also has the unique option to EFFECTIVELY proxy unlike most other champions who would struggle heavily to proxy due to kit limitations. This leads to one of two things:
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as described before, you two just farm out knowing that you can't do shit to each other and singed just scales to his gates-breach/50000MPH teamfight flip initiations with RG
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if you ever manage to kill singed in lane and try to push an advantage he'll just proxy you, which leads to either you trying to kill him and missing too many waves to feel good about it/you and your jg having to try and deal with him which means you have singed drawing so much pressure that it doesn't really matter if he dies because the map belongs to his team. Any decent singed will also have some wards when they proxy so that they can just execute if the jungler tries to deal with him.
Singed, in his own unique way, simply disallows any melee champion from being able to meaningfully interact with him. Yes his numbers probably would justify a small nerf, but to proclaim singed is BROKEN is a complete and utter exaggeration. To proclaim that singed auto wins melee matchups is MOSTLY true but very technically FALSE as he loses the darius matchup slightly according to statistics (and my own experience in the boostated land of diamond) and he loses the talon matchup MASSIVELY (understandable since talon has unique mobility which allows him to fuck with singed's proxy capabilities and an insanely strong cheese all-in in the early game against melee champs and especially singed since he can jump back onto singed if he gets flipped).
It'd be pretty hard to change singed to be healthier in melee matchups while keeping his kit relatively intact. Removing the grounded effect from his slow and compensating it by giving a stronger slow (basically a revert) could be a promising idea so that mobile melee champions can have meaningful counterplay against him.
Outside of massively reworking and giving singed an entirely new kit and identity, immobile melees will never be able to beat him outside of darius who has a few tricks to allow him to stick to singed relatively well when running ghost flash (which so many darii run).