What follows consists of HIGHLY opinionated information, but based on some significant experience.
When I was growing up in Cleveland, there were only three mouthpieces being played, the Portnoy BP02, the Gigliotti, and another one that I can't remember just now. A few really professional players played on custom Kaspars and Chedivilles, but for the rest of us who were just barely touching the professional level, those three were about it.
Now there are virtually hundreds of really good mouthpieces, from inexpensive ones to some over $700. Custom, hand-faced mouthpieces abound, but some really fine mouthpieces are made by machine. (The new thing is 3D printing, and I have two printed mouthpieces that are prime, made by Walter Grabner at clarinetxpress.com.)
Machine-made mouthpieces are generally very consistent, but any two custom mouthpieces will likely play differently.
I might as well get this out of the way. There are a few readily available mouthpieces that I've played that definitely don't recommend.
Any mouthpiece that comes with a beginner or intermediate horn. (Few makers provide a mouthpiece with their pro-level horns because they assume that the buyer will be using whatever one they choose on their own.) Mouthpieces that come with instruments are usually designed to be free-blowing and easier for a beginner to play from the start.
The Yamaha 4C, which is pretty much the same thing as those beginner mouthpieces
Any modern Selmer mouthpiece
If you have one of these, either pitch it or try another mouthpiece to see if it plays better for you.
What you should look for when searching for a mouthpiece:
Does it play well for you after a few weeks of playing it? (One minute of playing will only show up the major characteristics and doesn't let your embouchure adjust to something that might be a lot different than what you already have.)
Forget mouthpiece spec. tables, mouthpiece selection "guides,"descriptions, or price. You shouldn't try to predict how a mouthpiece will play for you by looking at specs, and a $50 mouthpiece might play better than for a thousand bucks.
Here are a few moderately priced off-the-shelf mouthpieces that I've played and that I recommend for anyone playing up to the late college level:
Fobes "Debut", a very inexpensive mouthpiece that has no faults
Vandoren 11-6, M15, M30, or 5RV, each with its own playing characteristics, but generally good. (I play the 11-6 as the alternate to my main one.)
Hite "Premier", also very inexpensive
I would avoid the Vandoren B40 and B45, which are very popular with students and a handful of pros. At first playing, they seem nice, but their tip opening is pretty huge, which makes control a problem for most players.
I would like to try D'Addario mouthpieces, but I don't have the need right now.
I wouldn't hesitate to buy only one mouthpiece from D'Addario, Vandoren, Fobes, or Hite because the are completely machine made, with no hand work done to them as far as I know, so each individual mouthpiece should be the same as all others of a model.
Well, I intended to go on in some detail about mouthpiece geometry, but I won't. My basic recommendations, especially touting the quality of the Fobes "Debut", are just results of my own experiences with only a few dozen mouthpieces over the years. Playing is everything, specs don't mean much once you blow.
Here's what you should do if you're looking for a new mouthpiece:
Play a bunch of them, not only different models, but different examples of each model if you're looking at"custom" mouthpieces! If the mouthpiece sounds good, has some resistance to it, and articulates decently, consider it a candidate.
Stay away from mouthpieces over about $300. (One custom maker charges over $900 for one of his models! I've played several of them, and never cared for them. Maybe others think the price is worth it, but I sure didn't. I prefer the $36 Fobes Debut over any of his that I've played.) Anyway, don't confuse price with worth; in my experience, as with clarinets and other accessories, there is little correlation.
Stay away from Ebay mouthpieces unless you can confirm that the seller is actually a player or a major maker and has excellent ratings!
Don't fall for "vintage" mouthpieces by legendary makers, Chedeville or Kaspar! They generally need refacing, and any refacing changes the mouthpiece from the original. A mouthpiece cannot be refaced to it's original spec.! They're generally priced at a ridiculous level anyway for no reason other than that they were made by legends. There are many excellent copies out there that have the same specs. as the vintage examples when they were made, and haven't been worn down so they need refacing.
Over and out.