I've got one that's doing the ol' dribble for a few minutes after flush. Ran out to the store today to get new seals with the plan of replacing the inlet seals and scraping out the gunk. However, it appears you can't buy just seals any more. Everything is a complete inlet side replacement. With how cheap they are, no surprise, the seals used to cost nearly that much! However, other than a couple that listed very specific toilets they say "fit almost all toilets"... Is this one of those things you just have to buy, try, and throw away if it doesn't fit? Other than carrying the old parts into the store and trying to look at them for comparison, is there a standard?
There are a few odd flapper styles, such as American Standard (read non-standard)
Others offer flapper and seat combo
Then there are simply universal flappers of various designs
Steps to repair:
Most recent flapper attachment mechanisms are plastic rings (old style were brass with bolts) that snap around the inlet tube.
SUPPLEMENT: Based on the added comments, it sounds as if your fill valve is clogged or broken. The valve may look like this (a relatively recent type)
The arm on top rises with the float around the tube and shuts off a valve internally. On this type, the top cap can be removed by pressing down on it and rotating counterclockwise. There is often grit in the valve, right under the cap, that can interfere with its closure. You can clean out the grit, reinstall the cap, and the valve may work fine. If not, replace it for less than $10.
There are other types of fill valves, including some older brass ones that may be harder to service. It is usually easier to replace them than fix them. They are pretty standard, height being the primary difference, and there are adjustable height valves available. They install pretty easily, but be sure to check for leaks from the bottom of the tank after installation.
There are a few newer, more complicated filling and flushing mechanisms, but you would need to provide a picture for info as to how to service.
From my experience, most toilets with a separate tank and base that you find in a home are standard. The few exceptions I've seen are:
I had to replace the lever in my Toto toilet. I went to the store, asked for a lever with a dual 45 degree bend "S" in the lever. The guy got me one. I had to return it, because Toto has multiple lever models with dual 45 degree S bends in them. I never thought they were that specialized.
In short, it depends.