Which gets you first when you are falling into a black hole, the black hole singularity or the cosmic background radiation? If you look up while you are falling into a black hole you see the universe blue shifted, that is, you see the universe moving quickly forward in time compared to your local time. Since this effect increases as you get closer to the singularity, a star aimed at you could blast you with a billion years of radiation in maybe a second. Of course, no star will be perfectly aligned for billions of years. On the other hand, there is lots of cosmic background radiation coming at you from everywhere in the sky and it will be blue shifted too. So, how bad is it? Would this radiation cook you before you hit the singularity?
 A: That is an incorrect description of what happens.
If the black hole is small, the gravitational forces pull you apart before you get anywhere near the horizon. On the other hand, if the black hole is large, you won't even notice anything as you cross the horizon, until you hit the singularity and then you'll be pulled apart. Remember, from your point of view you are just free-falling, the event horizon only has meaning for the outside observers. The stars will look somewhat different during your fall because of the hole's gravitational lensing effect, but no fiery death by cosmic radiation.
From the outside observer's point of view this is all different: they will see you getting closer and closer to the horizon but never reaching it.
Some more detailed explanation about this (and many other things about black holes) is for example here:
http://cosmology.berkeley.edu/Education/BHfaq.html#q3
A: I agree with Greg. There is no infinite blueshift, because you as an observer are also getting blue-shifted. In fact, due to the tidal forces, the radiation coming just behind you actually appears redshifted (although not infinitely so).
A: I remember calculating the amount of time one would take to fall radially inward with zero energy (so as to maximize the time taken) starting at the schwarzschild radius of a black hole of n solar masses to be 
$$ n(1.55 \times 10^{-5} ) \text{ seconds}$$
And you would never be able to tell that you had crossed it in the first place, the event horizon is simply a coordinate singularity (the Ricci scalar is very well behaved at the event horizon indicating large but finite curvature).
But since the curvature isn't infinite it doesn't make sense to say things are blue shifted to infinity for an inward falling observer, the only place where the curvature diverges is at the singularity and there isn't any understanding at that point.
Now you would have a problem if you attempted to remain stationary just outside the event horizon.
A: Time dilation and the blue shifting of incoming radiation go to infinity as you approach the event horizon, and you'll cross that before you get near the singularity. This means that, as you said, you'll be fried by blue-shifted radiation from the rest of the universe before your vaporized remains cross the event horizon.
Edit:
To clarify my answer, from the point of view of an observer approaching the event horizon of a black hole, down-falling radiation from a distant source will be blue-shifted. This blue-shifting goes to infinity as the observer approaches the event horizon.
Furthermore, from his point of view, the regions of the universe far from the black hole will be accelerated in time. The rest of the universe sees him slow down, and he sees the rest of the universe speed up. This time dilation effect also goes to infinity as he approaches the event horizon.
It is not really correct to say that an observer won't know that he is crossing an event horizon. The size of a black hole only affects tidal forces; they are small at the event horizon of a large black hole. But other effects are still important.
