The joint where my exhaust pipe meets my muffler unexpectedly rusted completely through, and my muffler literally fell off. (The rusty piece of crap is in my trunk right now.)
The car seems to run okay without it – and surprisingly not very loudly. [Happy face here.]
Until I get enough time and cash to replace the muffler, is the car safe and okay to drive? (FYI, it’s a 1991 Ford Tempo.)
Probably…with a caveat: the exhaust is now free to collect under the vehicle and possibly work its way into the passenger compartment. So, don’t sit in the car with the engine running for very long, and keep a window at least partly open. You’ll also probably see reduced fuel mileage, since the backpressure of the exhaust has been reduced, compromising efficiency (which was desinged around a certain amount of backpressure).
No, not really. You have a greater chance of carbon monoxide inflitration into the car. Obviously if you have an old beater with lots of rust, you have a bigger problem, but it’s not recommended.
The muffler’s sole purpose is to reduce noise from the engine. The exhaust manifold probably dampens the explosions that naturally come from the piston chamber but as far as the function of the car (or truck) itself, it will not affect it. It might even improve it a bit by allowing it to “breathe” easier without having to pump its exhaust thru the muffler.
However, in certain cities, you can be cited by the police for having no muffler for certain noise violations. Exhaust emmanating from the underside of your vehicle might attract unwanted attention. Not very loud to you might be excessively loud to another person. It does get much louder as you put on more gas. In california, a catalytic converter is mandatory and if it fell off too then you may be cited as a gross polluter.
My experience was the opposite. My muffler fell off, and I was unable to get it repaired for about six months. My gas mileage improved 10-15%, then returned to normal after I installed the the new muffler.
Don’t expect to pass your emissions check. 
Without the catalytic converter you are probably polluting as much as 10 cars. But as long as you are OK with that…
For all of the above reasons, I’d take out a low to no interest loan from someone you know and get it fixed pronto. I’ll leave the issue of whether or not to repay the loan alone.
YMMV.
Carbon monoxide is tasteless, colorless, odorless, and deadly.
You didn’t say where you are located, but in the Nothern part of the world it is winter and you are probably driving with your windows up. CO can and will kill you if the concentration gets high enough.
Based on what you have posted, I do not consider the car safe to drive.
I would suggest that you make the attempt to get it fixed ASAP.
On a Tempo, the muffler is the last part of the exhaust sysrtem. OK, a very short downspout is dead last.
What you still have left is a long pipe with a flange hooked up to the cat. converter. Emmision wise the car is fine. If it
would pass a “sniff” test before, it will now.
When you do decide to get it fixed, go whole hog and replace the pipe, gasket, muffler, and clamp. A WAG on the cost of parts only, from a parts store would be US$ 65-80. It is on of the easyer cars to replace an exhaust on. With a torch and jackstands, I have done it in less than 45 min. On a hoist >20. Good luck to you.
Pressure wave exhaust extraction is typically not impacted appreciably by a stock muffler, and is more a function of the diameter and length of the individual exhaust runners.
Maybe, maybe not. This is impossible to say without knowing the current exhaust setup and car model.
In California a catalytic converter is not mandatory for all cars. If the car is old enough, it may be exempt. In the case of the OP, it is mandatory.
He didn’t say his catalyst fell off - just his muffler. 
Based solely on the fact that his muffler fell off, you don’t think the car is safe to drive? With no other evidence at all? I mean, it’s not the best thing to do from a general standpoint, but you can’t make the judgement that the car is not safe to drive from the evidence provided.
LOTS of cars are on the road with holes in their exhaust, from leaky/missing manifold gaskets to holes in the manifold to broken EGR valves and piping to loose catalyst joints to holed catalysts to broken/holed/missing mufflers to broken/missing/holed pipes. When I was a mechanic, I saw (and fixed) tons of these. And saw tons of cases where they were never fixed.
However, as Booker57 says, getting a new muffler is cheap and easy, and you should do it anyways for the sake of taking care of your car, reducing engine noise, and preventing the small possibility of additional carbon monoxide getting into the car.
Erm I might be mistaken here, but isn’t the catalytic converter inside the muffler? I’m probably just mistaken. Can someone clear me up?
Another possible hazard : fire. The exhaust gas coming out of there can be HOT. I actually had an old foam-filled bumper burst into flame from exhaust gas (and, let’s be realistic, a bit of hot oil blow-by) on my old Toyota.
On most cars I’ve seen the underside of, the cat is pretty near the manifold. If I’m not mistaken, it’s because the catalytic reaction works better at higher temps. I’ve never seen a combination muffler/cat.
You are correct, the catalyst is always (well, 99.9% of the time) completely separate from the muffler. Sometimes, there are two catalysts (one near or connected directly to the exhaust manifold, another about a meter on down-pipe) but the muffler still typically comes after that. You need those high temperatures, in the absence of a reagent (like ammonia or urea) to affect the change from NOx -> N2 + O2.
There are a couple of reasons that they are separate, but one that I personally know of is cost - the catalyst body (referring to the steel, not the bed media or honeycomb) is made of a bit more expensive metal, to better withstand corrosion at the high temperatures, than the muffler is.
Mufflers aren’t very expensive if you do it yourself, you proabably do need new clamp(s). Shops charge a pretty penny to do it. But think of how easy you have it, just bring in the old muffler to an auto parts store & they can match it up.
Nope…the converter(s) is ahead of the muffler(s).
A popular performance package is called a ‘cat back’ system. You can replace all the pipe and mufflers behind the converter(s) with a hi-perf set up. 50 state legal for emissions.
You can even replace the cats with hi flow cats.
layout:
engine
engine exhaust heads
exhaust manifold/header pipes
to cat convertor(s)
to pipe
to muffler(s)
to exit pipes
to tips