What are Haskell's strictness points?

A good place to start is by understanding this paper: A Natural Semantics for Lazy Evalution (Launchbury). That will tell you when expressions are evaluated for a small language similar to GHC's Core. Then the remaining question is how to map full Haskell to Core, and most of that translation is given by the Haskell report itself. In GHC we call this process "desugaring", because it removes syntactic sugar.

Well, that's not the whole story, because GHC includes a whole raft of optimisations between desugaring and code generation, and many of these transformations will rearrange the Core so that things get evaluated at different times (strictness analysis in particular will cause things to be evaluated earlier). So to really understand how your program will be evaluated, you need to look at the Core produced by GHC.

Perhaps this answer seems a bit abstract to you (I didn't specifically mention bang patterns or seq), but you asked for something precise, and this is about the best we can do.

I would probably recast this question as, Under what circumstances will Haskell evaluate an expression? (Perhaps tack on a "to weak head normal form.")

To a first approximation, we can specify this as follows:

• Executing IO actions will evaluate any expressions that they “need.” (So you need to know if the IO action is executed, e.g. it's name is main, or it is called from main AND you need to know what the action needs.)
• An expression that is being evaluated (hey, that's a recursive definition!) will evaluate any expressions it needs.

From your intuitive list, main and IO actions fall into the first category, and seq and pattern matching fall into the second category. But I think that the first category is more in line with your idea of "strictness point", because that is in fact how we cause evaluation in Haskell to become observable effects for users.

Giving all of the details specifically is a large task, since Haskell is a large language. It's also quite subtle, because Concurrent Haskell may evaluate things speculatively, even though we end up not using the result in the end: this is a third breed of things that cause evaluation. The second category is quite well studied: you want to look at the strictness of the functions involved. The first category too can be thought to be a sort of "strictness", though this is a little dodgy because evaluate x and seq x $ return () are actually different things! You can treat it properly if you give some sort of semantics to the IO monad (explicitly passing a RealWorld# token works for simple cases), but I don't know if there's a name for this sort of stratified strictness analysis in general.

C has the concept of sequence points, which are guarantees for particular operations that one operand will be evaluated before the other. I think that's the closest existing concept, but the essentially equivalent term strictness point (or possibly force point) is more in line with Haskell thinking.

In practice Haskell is not a purely lazy language: for instance pattern matching is usually strict (So trying a pattern match forces evaluation to happen at least far enough to accept or reject the match.

…

Programmers can also use the seq primitive to force an expression to evaluate regardless of whether the result will ever be used.

$! is defined in terms of seq.

—Lazy vs. non-strict.

So your thinking about !/$! and seq is essentially right, but pattern matching is subject to subtler rules. You can always use ~ to force lazy pattern matching, of course. An interesting point from that same article:

The strictness analyzer also looks for cases where sub-expressions are always required by the outer expression, and converts those into eager evaluation. It can do this because the semantics (in terms of "bottom") don't change.

Let's continue down the rabbit hole and look at the docs for optimisations performed by GHC:

Strictness analysis is a process by which GHC attempts to determine, at compile-time, which data definitely will 'always be needed'. GHC can then build code to just calculate such data, rather than the normal (higher overhead) process for storing up the calculation and executing it later.

—GHC Optimisations: Strictness Analysis.

In other words, strict code may be generated anywhere as an optimisation, because creating thunks is unnecessarily expensive when the data will always be needed (and/or may only be used once).

…no more evaluation can be performed on the value; it is said to be in normal form. If we are at any of the intermediate steps so that we've performed at least some evaluation on a value, it is in weak head normal form (WHNF). (There is also a 'head normal form', but it's not used in Haskell.) Fully evaluating something in WHNF reduces it to something in normal form…

—Wikibooks Haskell: Laziness

(A term is in head normal form if there is no beta-redex in head position¹. A redex is a head redex if it is preceded only by lambda abstractors of non-redexes ².) So when you start to force a thunk, you're working in WHNF; when there are no more thunks left to force, you're in normal form. Another interesting point:

…if at some point we needed to, say, print z out to the user, we'd need to fully evaluate it…

Which naturally implies that, indeed, any IO action performed from main does force evaluation, which should be obvious considering that Haskell programs do, in fact, do things. Anything that needs to go through the sequence defined in main must be in normal form and is therefore subject to strict evaluation.

C. A. McCann got it right in the comments, though: the only thing that's special about main is that main is defined as special; pattern matching on the constructor is sufficient to ensure the sequence imposed by the IO monad. In that respect only seq and pattern-matching are fundamental.