How to reproduce that low-quality, noise-filled distorted effect of a badly tuned radio?
What I need to know is both what *type* of effect to use (for example equalizer, compression, gating etc.) *and* some good brand names. But nothing Sonar-only - I want to be able to use it in other DAW's, too.
 
Thanks.

A sound design question eh?
 
You have already described what you need to do.
 
Sonar......A Parametric EQ shelved off correctly (Low and High) in the right spots or just search the sweet spots on Quad curve, some boost on the mid range and either a touch of gain or/and some distortion, TH2 would do the job with some Amp Simulation, also some tape emulation, some concrete limiter and a noise gate.

I do this effect all the time but no way will I reveal my sound design tricks on a public forum 

I would combine:

1. EQ (hp & lp)
2. ring modulator
3. amp sim

Forget about brand names. Generic stuff will be fine.

1-Sonar has FX-chains in its browser called Vox - Vocal Comb and Vox - Telephone Delay.  You can drag one of those into the FX bin and then play with the settings to get what you need.
2-EZ Mix also has a number of choices.  (Sorry, I don't remember which are in the standard set and which are in the Vocal Toolbox add-on).  These sound great.  "Crappy TV" and "Cosmonaut" are especially good.
 
From you description, I think you want an extreme midrange boost and an extreme cut of highs and lows.  You can clone the track, reverse phase on one track (which will effectively result in silence), and then on one track only, boost the midrange.  The result will be that you will have nothing except the boosted range.  You can move the boosted frequency around to find the sound you want.
 
For the static, you might look online for a radio static sound effect and add it on.  I download a lot of sound effects.  Some are free and some just cost a couple of dollars.
 
 

Voxengo Voxformer has a "telephone" setting.

The main ingredient is high/low filer cuts.  As noted, there are various others things 'that can help - a hint of distortion.  A gate if you want a squelched sound (police, NASA, air traffic, military radios use this).  You can use phase to "lose" the signal.  Short delay to create ring.
 
Lots of fun.
 
@

2 bands did 21st Century Schizoid Man : same engineer on both : The box used was a Big Muff !
 
Ok there were more than 2 recordings, but you would only remember 2...
 
 

The free iZotope Vinyl plugin will do it too:
http://www.izotope.com/products/audio/vinyl/
 

... what they said.

From memory bifilter 2.