Sound deadening on the cheap!

You could even consider just painting the underneath portion of the sill (only). This would not be visible but it would still reduce noise. I find the sills make quite a bit of noise when dirt is kicked up from the font wheels and resonates when it hits the underside of the front sills. That said, I guess one could also opt for front mud flaps for the same effect. 
There are also internal frame sprays that might have the same effect from the inside. Not sure, never tried this. There are access holes for spraying into the front sills in the front wheel wells behind the plastic wheel guard. For example:
Eastwood Internal Frame Anti-Rust Coating, 14 oz. Spray
 

I am planning on treating the underside and inside the sill panels with Bilt Hamber spray as soon as I can. 
The underside is imaculate and I want it to stay that way. I need to lay it up in my garage to do a really good job but my closet Queen MX5 resides in there just now until it sells. I daren't put it outside and let it get dirty or wet! It is a problem I brought upon myself.
I will be doing the boot and the iner wings from the air vents with bitumen tape at the same time.

Just had a wonderful ride in my little car around my country roads circuit of about 15 miles with a lovely quiet and rattle free chassis.
That's all, nothing further to report..except..did I ever tell you I love this car?

Inspired by your comprehensive application of bituminous padding & having already done the more resonant panels, I went around the car tapping panels to see which “rang”.
I’ve done a bit more, easily subduing clangy bodywork & earmarking those to be treated.
Only when out of hibernation will it’s effectivity be noticed; it can’t make it worse.
M

Good on ya mate!
By the time you take her out you will have forgotten about what you did but may be pleasantly surprised at how refined she feels. 
It has genuinely transformed mine. 

So today i was removing my door cards in order to swap out the "technical" trim pieces for some i'd sprayed yellow (no not Sunspot but as its the interior i don't think it's as important with there being no direct comparison, i'm sure others will disagree and they're welcome to send me a few litres of Sunspot to solve the problem) while i was in there i thought "let's get this sound deadening done" so that's whta i did and i thought i'd show my work here so you can tell me if i did it right or not.
I wasn't able to go between the crash bars and the outer skin as the adhesive/foam used by the factory went all the way down the length of the bars but i did manage to get everything pretty much covered. Haven't taken the car out yet so can't say anything about any changes to driving rattles but i didn't notice a great deal of difference in terms of the noise of closing the door so not sure if i've done it wrong? i even listened for a difference between the two doors after finishing the first one. At the very least though i'd imagine its good for the general insulation of the car and hopefully i'll notice more of a difference on the road. Plenty left over so might do the bonnet and boot next.

TA22 has taken a simple idea & moved it to a level, never seriously considered. Inexpensive sound deadening is almost an oxymoron & I doubt you can go far wrong with this approach. You can’t have failed to suppress some, if not all the panel resonances, certainly the closing of my doors is a more subdued affair than previous.
I started messing about with this stuff quite a while ago, resulting in my being used to the amended sound. Sometimes I can’t remember what I had for breakfast, let alone the sound of my doors being shut several months ago.
TA22 comments carry much greater weight than mine, I’m certain he will be along shortly to add some meaningful additions.
M

Hiya Kid..
I got the idea from AP and ran with it.
What you have done looks fine and I couldn't get mine between the bar and the doorskin either so I just wrapped it.
There isn't a significant difference just closing the doors but on the road it stops a lot of resonance. The panels don't zing anymore.
You saw my post on it so you can see how far I took it. The floorpan and behind the seats make a difference. It's hard to believe there is nothing on the floorpan from factory.
I have yet to do the boot lid and the rear panels in the boot but I soon will.

We were coming back up the M6 from Keswick yesterday and my wifes car shed a pad which knackered the disc so today I went in to Aberdeen to ECP for discs and pads. I haven't been out in the TF for a few weeks so I decided to go in that and in all honesty I was surprised how quiet it now was. It really felt quite refined and "solid" on the road. I was smiling the whole way!
It is well worth the effort and keeping a roll handy in case you suddenly do a job that exposes a bare panel is a good idea.
Good luck.

Hi all

So far I have "treated" thoroughly every bare part I could reach (including from the lateral vents), bonnet, boot (upholstering inside the lip hollow made a lot of difference), floorpans and engine cover, getting rid of the dead dog in the process. That is, everything EXCEPT the doors (next task)

It not only has improved dramatically the whole feel of the car on the road, but also the doors close in a very solid way, thump instead of zing. What means that what matters is deadening the general resonance of the car, rather than particular isolated parts. That would explain KidCreation´s modest outcome after treating the doors only.

That's good to hear. My car had zero sound deadoning from the factory..not even a token sticky pad anywhere and so it stands to reason that anything we do has to improve the car.
Your doors will be the finishing touch I think.

I think so. I've found sound deadening to be a great improvement for this car, and I've got to know it better as well in the process.
Doors treatment will follow, along with tweeters/speakers upgrading and Superlock suppressing as per Notanumber instructions.

This Forum can change completely one of this cars. So much better than when they left the factory 

I agree that doing it allowed me to learn more about how the car is built..where cables and looms are run etc.
I have not come up with anything original to add to the Forum because everything I have done has come from the Forum.
I added tweeters too and upgraded the main door speakers at the same time. Still got the T bar ones to replace.