Sounds like a sort of nightmare. Red hot splinters forcefully thrown injuring people and setting their surroundings on fire...
Have you experienced something like this? What was it like? What kind of lamp it was and what was the most probable explosion cause?

I was near to get an explosion two times:
1) When i was child, i mistook an connection of two pll lamps and i bypassed ballast; an big flash, some noise of thermal expansion and no sign of live on bulbs.
2) When i was in an market, one of halogen lamps exposed was arcing, but with no conseguences.

Incandescent lamps in my hostel, made a boom sound and a bright white flash when EOL, and tripped the main circuit breaker, but didn't exploded or came off the base.
In my mother apartment, we had GE 40W and 60W krypton filled mushroom incandescent lamps, that ejected from their bases at EOL without explosion noise, and without a big flash.

Once I screwed in a mercury bulb, thinking it was a self-ballasted one. It was a standard mercury bulb and I didn't switch off the fixture, so the internal quartz bulb exploded while the outer glass bulb didn't.

Those stupid U shaped fluorescent tubes have exploded in my hands numerous times while changing em out 

I've also had a metal halide bulb blow up within seconds of screwing it in.  Thankfully I had gloves on and was closing the cover at the time. 

Last year I had a C7 Christmas lamp break at EOL. No particularly bright flash, but a definite pop sound. I also had a clear C7 night light lamp that went EOL (normally, not violently) when I was about 12 years old. I shook it until a small portion of the filament was directly between the lead in wires (making it about 1/6 its normal length). I carefully screwed it back into the nightlight and turned it on. I had a very bright bluish flash, and the inside of the lamp was coated in an almost mirror silver coating, probably vapourised metal.

It was when I was a kid living with my parents, the incandescents there tend to all the time explode at their EOL. Sending the glass pieces across the complete room. It happened all the time, regardless of the lamp brand. And what was really annoying, the main circuit breaker usually gave up after about 3..5 lamps...
 A relieve came in 90's with the first PL-S available here (any longer fixture was converted to the 11W PL-S) and then integrated CFL's for the rest...
Since that time I do not trust safety of any incandescent unless it is in an enclosed, robust fixture clearly able to contain such explosion...

@Cavannus: Once I bypassed the ballast of a coated MV lamp by mistake. There was a big flash but - surprisingly - the lamp has survived! The breaker (although a quite old one) was faster.

@Medved: I think I saw an incandescent explode once or twice (or the glass bulb just fell off) but this amount is rather surprising. Didn't you have overvoltage at your parents' ...?

I think (extra/) high pressure lamps, especially the bigger ones where the hot arc tube lights by incandescence literally for tens of seconds after turning off will probably be the worst. Smithereens outside the outer bulb will probably cool down much faster but still could be really dangerous. Fortunately, I've never experienced anything like this.

When I was 6 years old, I wired a 6 volt mini Christmas light bulb straight to 120v, it made a loud pop and shot out of the socket like a gun 

@Medved: I think I saw an incandescent explode once or twice (or the glass bulb just fell off) but this amount is rather surprising. Didn't you have overvoltage at your parents' ...?

Not overvoltage (there was ~225..235V all the time, so in the upper limit of the tolerance in that era, but bulbs rated for 240V), but very low impedance of the mains connection (the housing block have 8 floors, 4 dwelling units on each, the electricity was fed via four ~100x5mm aluminum stripes, cable to the transformer is still way thicker, transformer is barely 200m away), so the short circuit currents are really high.
Where living there I didn't know the mains voltage could ever sag more than a fraction of a volt even when turning ON a 10A load...
Of course, that means short circuit currents in higher kA's, so when created by an arc in the lamp, the arc becomes really fat, so the bulb just could not contain that...

And an EOL arcing inside of a gas filled incandescent is just not avoidable above ~30V, normally it should be handled by the fuses inside of the lamp. But each fuse has some it's maximum current limit, which could break. When going above, the fuse disappears, but "get replaced" by another arc and the current is still flowing...
On top of that, with higher currents it takes longer time to quench the arc between contacts of the circuit breaker, so it does not help either.

I once wired up a ballast wrong and a Philips 11w PL-S lamp was powered directly from the mains, I don't think anything other than a bright flash from the electrodes happened though

I have never actually had a lamp explode shooting shards of glass everywhere but I have inadvertently put 12 volt bulbs in 120 volt sockets causing a loud boom and bright white flash.. Also one time I had a 60 watt soft white GE go with a loud bang when I turned the switch on.  Another time I had a 60 watt GE Reveal bulb that must have had a slow air leak...it went out a couple minutes after being screwed in for the first time and it was all smoked up on the inside. 
As a 4 year old or somewhere thereabouts I stupidly tried putting 2 candelabra base nightlight bulbs in a regular table lamp socket and got a shower of sparks.  I ran downstairs, and my mom  could not figure out WHAT I was running from!

Well,  I had a F15T8 lamp explode on me!  But that was long ago when I was 9 or 10 years old.   Took apart a cheap Lights of America undercabinet fixture and in a rush to wire everything back up on a piece of wood, I accidentally bypassed the ballast.  When I plugged it in,  saw a bright flash, and one end exploded.   And of course,  blew a 15A fuse!  My father ended up stockpiling on fuses!

2-3 times I have seen one.

We have a ceiling fan with a light dimmer and fan speed control installed with a wireless wall mounted remote.  Years ago we were playing on our GameCube hacing fun, and one of the flame shaped incandescent bulbs decided it wanted to explode and something landed in my hair.   I don't remember what because it was so long ago.  The light dimmer was messed up after that and only did 2 light levels (used to do 100%-0%), it doesn't work at all now.

Another time an incandescent light in the bathroom exploded for no good reason.

And then I blew the cathode in an F40 T12 because I grounded it to the troffer housing with it turned on (that was smart).

I was installing a new Sylvania 25-watt candelabra globe lamp in one of my hallway lights with the power on. As soon as the lamp received power, there was a loud pop and the glass envelope broke. Luckily I didn't get cut, but it sure scared me!