Having been watching Bad Game Hall of Fame's streams, a question that has been bugging me for weeks now, which console group has the worse games, avoiding obvious failures like 3DO, Phillips CDI and Atari, were just focusing on the big four, Sega, Nintendo, PlayStation and XBox.

Sega
We'll start with the group no longer making consoles Sega, having not been around for 20 plus years now, there isn't much to work with and Dreamcast's library has hardly any bad titles, however with the Genesis going on so long it produced some quite bad titles, Mega CD and 32X being particularly bad with games, surprisingly the Saturn has very few truly bad titles but doesn't have the best track record adapting it's Sega arcade catalogue.
XBox
I've heard this a million times now, achievement hunters only play bad games, I've seen your profile; look almost everything Xbox has ever had is available on the other consoles, in fact alot of companies try and avoid porting to Xbox entirely as it's proven to be harder to port on xbox especially the more recent ones.
PlayStation
With PlayStation having a relatively good track record between all it's consoles, it's hard to imagine it having a bad game catalogue, however it's because of it's over whelming success that everyone tried to port on to it for a quick buck with very little push back from Sony themselves not to mention it has xbox's so called cheap achievement games.
Nintendo
You'd think Nintendo wouldn't have this burden with it's immaculate record on it's first party titles but that's the thing, it's first party record may be immaculate but it has a porting problem, Nintendo is so keen to go it's own way that anyone wanting to port on it has to jump through more hoops more than Xbox, it's not like Nintendo isn't capable of making something better, Gamecube is proof of that but also like the PlayStation, something too popular can create an endless supply of shovelware titles which were still feeling now thanks to the Wii.
Final Verdict
Sega have been out the game too long to really have a bad catalogue with the majority coming from it's attempts at keeping the Genesis alive.
Xbox's catalogue is available on other consoles, if anything it's redundant in the grand scheme of things.
PlayStation has no quality control when it comes to it's catalogue and it has no excuses for allowing it to be that way.
Nintendo has a porting problem as it won't make something that caters for anything other than it's first party games.
Really it's between PlayStation and Nintendo, but I'll give it to PlayStation for one major reason. Corcord. Critical failures are a part of a consoles life cycle but only PlayStation has failed so badly that it has lost more money than all it's previous bad games combined.