What is the best way to clean Vinyl?

Yes, indeed @clearthinker the engineer working on the Les Paul tracks was using a Simon Yorke and there is a photo of some of his set up in this article: 

Lots of great knowledge and experience have been mentioned in these many responses to the problem of cleaning records. I'll add my method that's a bit time consuming, but effective, and stays well below the cost goal of $500. For the price of a Spin Clean, and maybe $10 in plastic, felt, and wood (plus a shop vac), I clean about 6 records an hour, with them completely and immediately dry afterwards.  I use a Spin Clean first, then vacuum it dry with a PCV wand made with a wood dowel for the center hole, a slit made with a dremel tool extending from just inside the label to just outside the record edge, and felt (I think) around the slit for the record to rest on. Attach a shop vacuum to the end of the wand (other end taped shut), and anything loose left behind by the Spin Clean (including the solution) comes right off. Care is needed to not drop or scratch the record while handling of course (almost happened a few times) but it is obviously effective as my wife complains she can't hear any clicks or pops, while I love how silently they play (usually can't hear the lead-in groove). Started just with the Spin Clean, but could only do two or three albums before the towels were all too wet to completely dry all the albums I wanted to clean that session. Just another alternative method to consider.

@whart 

Thanks for that.  I don't suppose you got to meet Simon.  He had installed that equipment many years previously.

Another vote for the Loricraft 4. Best vacuum pump going. And the new vinyl string each cleaning ensures a great cleaning. 

The least expensive vacuum unit is probably the way to go. There are numerous ways to put fluid on the record and brushes to use. It is drying the record and removing all the residue that is critical. Air or fan drying is a  bad idea. Regular distilled water is not no residue water . It is lower residue water. Put a puddle on clean glass and look what is left after it dries. Nobody is going to get laboratory water to use in their record cleaning fluid, too expensive. Many of the additives used in cleaning fluids do not evaporate, they have to be removed. 

Most of the noise on records is from contamination in recycled vinyl or just plain bad pressing. It will not clean out. Those of you who think even 50% of the records out there are noiseless must be listening at low volumes only. On a wild guess I would be lucky if 10% of my records are state of the art pressings.

On the bright side the only real danger to records is physical, scratches, heat and so forth. It is very hard to hurt vinyl with the fluids commonly available to consumers. I have exposed records to 99% alcohol, denatured alcohol, lacquer thinner, brake cleaning fluid, mineral spirits, Naphtha, and gasoline. None of these had any audible effect on the record. I soaked 5 analog production records in brake cleaning fluid for 24 hours. All five including the labels are in perfect shape and it has been over a year since I did it.

The cheapest cleaning fluid you can use effectively is one cup 99% isopropyl alcohol in a gallon of distilled water with three drops of Tergitol (a surfactant). The Tergitol functions as a wetting agent and keeps the water from beading up on the vinyl. If the water still beads up add a few more drops. Tergitol does not evaporate so you want to use as little as possible and should be used only with vacuum drying or you will see a wad of Tergitol form on your stylus. It won't hurt anything but you will have to clean your stylus after every play. The more you clean your stylus the higher the risk of you damaging it. One rum and soda will do it.