Can tin foil hats block anything? This is joked about all the time, but...
Can tin foil hats actually block anything? If they can, what frequencies? Is there any research into tin or aluminum foil and radio blocking or amplifying abilities when shaped into a hat?
If they really don't do anything, what would be better? Radio blocking in a hat design that is, not a Faraday cage suit (with matching tie).

Edit: people seem to be missing the tags, so I’ll clarify the questions – What frequencies (mainly on the radio range) can tin foil, when shaped into a hat, block or at least attenuate? If not, are there any frequencies that resonate/amplify within the hat? What would work better for signal blocking in this regard (to the head that is)?
Yes this is a funny subject but let's actually take a look at the physics. There are related questions to the hat, but I believe they fall outside the scope here. If someone wants to ask those on the exchange sites, go ahead.
 A: I'd like to point out that blocking UV radiation is useful, as I for one have had sunburn on my head. Foil could be used as an improvised "beenie" hat if stuck out in the open, or as a black-out lining of a hat that is more decorative than opaque (e.g. knit fabrics).
A: 
Can tin foil hats actually block anything?

Anything?
Sure.
As already noted by Daniel Griscom's answer, tin foil can block several "things" including rain, alpha rays, and electromagnetic radiation with small enough wave length that the radiation cannot diffract around the edges of the hat.

If they can, what frequencies?

Since you mention frequency let's really focus on electromagnetic radiation.
Foil, being a metal, is very good at reflecting electromagnetic radiation.
You can see this quite easily by looking at a sheet of aluminum/tin foil: it is shiny and you may even see a reflection of yourself.
Physically this happens because electrons can move easily within the foil.
Electromagnetic radiation impinging on the foil causes these electrons to move, and that motion produces new electromagnetic radiation which generates new radiation which destructively interferes with the original incoming radiation. 
This leads to cancellation of the incoming wave so that it does not penetrate the foil, and creation of a new outgoing wave which is the reflection you see.
Now this isn't the whole story.
Radiation with a wavelength at about the same size as the hat actually winds up resonating inside the hat.
In an effect similar to a vibration of a violin/guitar string, radiation that fits nicely inside the metal cavity formed by the hat actually winds up creating electromagnetic fields which are bigger than the incoming field.
You can get a rough idea of how this works on the Wikipedia page for resonance.
It's the same effect as when you blow gently over a bottle and hear a loud hum sound.

Is there any research into tin or aluminum foil and radio blocking or amplifying abilities when shaped into a hat?

Yes, there is a sort of famous study from MIT where a student looked into this.
They found the resonance effect mentioned above actually made the electromagnetic field strength inside that hats larger:

"For all helmets, we noticed a 30 db amplification at 2.6 Ghz and a 20 db amplification at 1.2 Ghz, regardless of the position of the antenna on the cranium. In addition, all helmets exhibited a marked 20 db attenuation at around 1.5 Ghz, with no significant attenuation beyond 10 db anywhere else."

Note that resonance is not the same thing as amplification.
Amplification means an increase in power; you can only get that if you have a power source, as in a radio receiver which amplifies the signal coming in over the air waves.
Resonance is an effect wherein (at least in this case) the incoming flow of energy winds up being sort of piled up into the hat so that the fields in the hat are larger than the fields coming in.
It doesn't add new energy to the system, it just concentrates it in the hat.

If they really don't do anything, what would be better? Radio blocking in a hat design that is, not a Faraday cage suit (with matching tie).

Radio waves have pretty big wave length.
For example, a 100 MHz wave has a length of 3 meters.
If you want to block that out you need something like a Faraday cage with holes a substantial fraction smaller than that wave length.
P.S. The conclusions in that MIT study are hilarious and I recommend reading it.
A: A tin foil hat can block:


*

*alpha rays

*electromagnetic waves, where the wavelength is short enough to not diffract around the edges (counterexample: FM radio waves), but not short enough to punch through the foil (counterexample: gamma rays)

*ultrasound

*rain

*job offers

*marriage proposals
But, then again...
A: Whereas a straight unfolded, uncrumbled piece of tin foil can block out some of the lower wave lengths permeating throughout the atmosphere, a folded or crumpled tin foil hat will not block out anything.
The reason is that tin (aluminium) foil is actually made up of two thin layers.  There is a clear sheet in between the layers to act as a base in the manufacturing process.  If folded or crumpled the aluminium (tin) foil breaks down and creates visible patches throughout the piece. 
Try crumpling and then restoring a piece of foil and then hold it up to your face to see the breakdown spots emerge.  
In this condition any wavelength would be able to penetrate the layers of foil making the idea of protection moot.
