Well, seeing the list of unusual happenings here, I'm inclined
to post a few of my own. By the way, I have been hanging around
here for sometime before I joined, so I do know some more "inside"
details in this forum, more or less.
Seeing the incidents about odd people on buses on the first few
pages made me want to write a few of my own. The first incident
happened on SBS1688K on route Express 518 (back when it was still a
perm. bus on that route), travelling in the direction from the city
to Pasir Ris. I'm not quite sure where this person boarded the bus,
but he was a rather odd man who wore a lot of weird things on
himself if I remember correctly. He talked about a lot of
philosophical things, including retribution and such. I can't
remember much, but I do remember thinking about how meaningful his
impromptu "speech" was. He alighted at either Bedok or Tampines
(I'm not too sure). Later, a female passenger asked the BC whether
that man was crazy, and the BC said something about that person
frequently "visiting" different buses, either to give his "speech"
or to sing a few songs. 
More recently, on bus 76 (a MB O405, not sure of the registration),
while travelling from Yio Chu Kang to Parkway Parade, when this
person boarded somewhere halfway through my journey. Before taking
his seat (right at the front of the bus, just behind the driver's
seat), he could blow his whistle loudly, once, which startled
probably everyone (including me). The BC, however, took it as
nothing happened. He stayed on the bus and did nothing else until
he alighted (somewhere around Upper Paya Lebar, I believe); he blew
his whistle again while exiting from the rear door, which again
slightly startled me.
p.s. Yes, I'm a Singaporean - I'm just more inclined to type
properly without Singlish...