How to attach this tap cover

I would venture a guess that @Tester101's suspicion about your floaties coming from your glass (unrinsed soap, dust, etc.) may be heading down the right path. I would try rinsing a glass, scrubbing it with a new sponge or clean washcloth, wiping it dry with a clean towel (not paper towel, as this may leave pieces of paper behind), and then refilling it. @NiallC's comment about asking for a water report from your municipal water supply is a pretty good alternative, but doesn't factor in anything that may come from the pipes between them and you. Most health departments will do water testing, so you could always fill a bottle and take it in.

Beyond that, if you really have to know what's in the water, your only real solution to absolutely know with 100% certainty (technically, unless you watch the health department do the tests, and can verify their results, you can't KNOW that they're right...but I'd probably just accept it) is to get testing kits for this, that, and the other thing (that was just the first page offering a wide range of kits that I found), that operate similar to what you might use for pool water. Depending on the sensitivity levels, you may pick up on things like trace amounts of chlorine...but of course this is going to get back into the more expensive end of things, as those kits aren't cheap (Grainger's seem to run around $40-$50 and up...it's about $20 for their cheapest, which is testing for copper, on that first page of results).

You could pick up a pool test kit, which will cover a couple of parameters (chlorine, calcium, hardness, etc.), but it's not going to cover everything and may not operate in the range you need to identify your floaties...it's designed primarily to help you maintain clear-looking pool water that you're not drinking, not crystal clear tap water.

So really:

• Ask for report from municipal water supply

• Take in sample to health department

• If you're still paranoid and want to know what it is, buy test kits

• If you just want to get rid of the floaties, filter, filter, filter, and make sure your dishes aren't the source

How many splines are on the broach, how long is it and what's its diameter?

As according to faucetpartsplus.com, that's all you need to get started. They have an entire list for spline counts, listing them (below) to specific manufacturers. If you can't figure it out, they suggest sending them a photo, as they're very interested in selling you parts; manufacturers and their employes might not be so inclined. It's more than likely they'd only be able to refer you to who makes or has the replacement parts these days; not their prerogative.

Even though the link goes to an excellent array of pictures and contains most of the leading brands, I implore you to contact your local parts suppliers who are usually more than willing to help you.

You mentioned plumbing suppliers. Those are places you buy pipe and fancy toilets, you need a repair part supply house.

4 point square broach - Milwaukee, Royal Brass, Speakman, Sterling, American standard, Chicago, Concinnity, Gerber, Price Pfister, Symmons, Zurn
8 point broach - Briggs
12 point .335" broach - Crane Dialeze, Michigan Brass, Wolverine.
12 point .375" broach - Arrowhead, Artistic Brass, Harcraft, Glauber, Price Pfister, Scoville, Universal Brass.
12 point .39" broach - Bradley, Elkay, Fisher, Sears, Universal Rundle.
12 point .415" broach - Symmons.
12 point .485" broach - Crane, Symmons.
15 point broach - Santec, Savoy, Wolverine.
16 point .360" broach - Gerber, Sayco.
16 point .370" broach - Acme, American Standard Cadet & Colony, Barnes, Burlington, Central Brass, Concinnity, Eljer, Glauber, Harden, Kohler Trand, Milwaukee, Newport, Phylrich, Royal Brass, Scoville.
16 point .40" broach - Sterling.
17 point broach - American Brass, Mansfield, Midcor, Phoenix, Streamway.
18 point broach - Briggs, Indiana Brass, Union Brass.
20 point .285" broach - Broadway, Concinnity, Danfoss, Dornbracht, Eljer, Grohe, Jado, Kohler, Milwaukee, Paul.
20 point .415" broach - Broadway, Speakman, T & S Brass.
22 point .375" broach - American Standard old 3/8".
22 point .438" broach - American Standard current 7/16".
38 point broach - Import, Pegasus, Glacier Bay.
D Broach - Delta, Milwaukee, Universal Rundle, Valley.
Oval Broach - Delta, Moen, Mixet.
_{faucetpartsplus.com}