I'm a Dummy, Tell Me About Turntable Mats

No felt. No tape. No weight. No problem.

@lewm, It is not really record noise. You are hearing the mechanical system of the record/stylus/cantilever/tonearm vibrating. Ideally, all the energy spent in moving the stylus should be translated into an electrical signal. Some is turned into heat and some is turned into noise. The noise is a problem because it reflects back on the structures creating it which you theoretically can hear amplified along with the music. 
Let us call this "tracking noise." Tracking noise has been mentioned by others as a measure of the quality of the playback system. Certainly, those old ceramic needles were very loud and I remember my fathers old Rec O Cut with its old Empire cartridge was easily audible 3-4 feet away.
My current combination is very quiet but my main point is that clamping the record properly makes an obvious improvement. 
I had a Transcriptors turntable at one point decades ago, the one that set the record on 6 brass weights. I hated it. It was pitiful in comparison to an LP12. I blamed the platter for that.
I'm sure Chakster and Rauliruegas can comment on this.

Mijo, You wrote, "It is not really record noise. You are hearing the mechanical system of the record/stylus/cantilever/tonearm vibrating."  How does that contradict anything I wrote?  Moreover, your statement itself is a bit contradictory internally (to say it is not "record noise", which was your term, not mine, and then define it as in part due to mechanical energy due to the record, along with the other factors you name).  Anyway, I view it as a "problem" endemic to vinyl reproduction, and it probably varies all over the place based on the tonearm design and construction and the cartridge design and construction, and I have better things to worry about. Is there any test or are there any measurements that correlate SQ with reduction of this "noise"?  In other words, if you believe it to be undesireable, what does one do to reduce it, and does that result in better SQ?

It’s called stylus jitter. https://youtu.be/F65mODzn4Gk?t=778Yes the problem is endemic. Yes it varies all over the place just like you said. I just bought a SG200 so hopefully will hear all the wonderful benefits PL talks about here.

A Turntable has many options on how to create a differing interface between the Stylus > LP > Platter, in which most cases a Platter Mat is the Option, even though there are some other methods used of which some users are convinced works for the better in their set ups and environment,
Focusing on a Platter Mat, there are a large variety of materials from Paper to Metal, as well as identical materials produced in a different thickness.
Each material type will be used to perform one function only, which is to create a Layer in a construction.
A Typical Construction being a Rack/Shelf > Footers > Sub Plinth > Footers > Plinth > with the Typical additional attachments being Turntable Chassis > Platter > Tonearm.
A Tonearm receives a Cartridge as the fastened attachment and the Platter usually receives a Platter Mat, which is most commonly laid on the Platter without any method the fasten it.

A Platter Mat can have a profound effect on a presentation, there are interfaces created using certain materials where the perception of the replay is immediately detected as being a detractor, and the loss of detail and smearing of information is quite noticeable.
There are also interfaces created using certain materials where the performance is detected as being an attractor, the information retrieval is quite impressive and there is also a perception of realism and honesty that are quite noticeable.
I am not sure if there is a ubiquitous mat material that offers the same properties in all set ups and environments, as I have had exceptional results with certain mat materials in my own system, and when these and other owned mat materials have been loaned into other systems the impressions were changed and other mat materials had been given favour. 
It is really a case of turnover of mat materials until the most impressive one is discovered.
In the case of myself and most people I know with a TT,  Rubber and Felt Mats are not usually maintained, they are exchanged for other mat types.