The subways are technically run by the Ministry of Public Works, who provide oil for the wheels and employ members of the Steamworkers' Union to take carriages off the line and check that nothing's going to go wrong. There is something of a problem starting to brew here. The Ministry is bloated and corrupt, so funding is continually being reduced. This means that the carriages are becoming less and less safe. Worse, Otto's overconfidence in his own genuis has resulted in a scarily low number of fail-safe mechanisms. When an accident does happen, it's going to be a biggie. So far there hasn't been a catastrophic disaster, but something's going to have to give sooner or later.
The situation is not helped by the fact that the entire subway system working so well for so long is a big slap in the face to the Clockworkers' Guild. To say that they're rather resentful of it would be an understatement: they have a vested interest in seeing it perform badly - or even suffer the aformentioned catastrophic disaster.
There's also the question of the Beggars. The Beggars have colonised the subways: they're technically public property, they're sheltered, and there are plenty of passers-by to beg for money. The subway carriages tend to double as sleeping quarters for members; even the King of the Beggars himself sleeps incognito there, protected by the fact that nobody outside the Beggars has a clue what he looks like. Besides, who pays any attention to a beggar on a subway?
They're also starting to colonise the rest of the subway tunnels. Not the ones with the trains running through them-- the ones which have been abandoned, or new tunnels which they're starting to dig themselves. They serve as handy crash space and safe-houses for when they need something secure, and they connect with the catacombs under the city in a number of places. They have an unparalelled knowledge of the subway system, surpassing even that of the engineers who work upon them.
Finally, there's the problem of what superstitious folk call 'ghost trains'. It's true that from time to time, some subway carriages go a bit AWOL. Nothing serious, they just suddenly take a vastly different route and careen to a completely different station. Nobody seems to be able to work out why they do this; currently it's being attributed to incompetent engineers.
[With regard to the many, *many* Shell Beach rumours, the station doesn't exist. It is a red herring. It was going to be a station but funding ran out, and they didn't have time to change all the maps. In fact, it only exists so that people who've seen Dark City can have a good giggle at the Rumours.]