One other thing -- we tend to accept negative coincidences that hurt the hero by making his dilemma worse, but rightfully reject those that help the protagonist.

It's the "at least nothing else can go wrong!" sudden thunderstorm cliche, which works because we all know what a bad coincidence feels like, so we can relate and accept it. But very few of us ever get a helpful deus ex machina in our daily lives.

Dickens was the master of using coincidence. He not only embraced these techniques, but had a philosophical position about surprise being a part of life, that we all should accept and enjoy.
Of course he was also a master of character, so we delight in his unexpected revelations.
In "On the Waterfront," it is a coincidence that Joey Doyle's sister,Edie,happens to be visiting her family,when her brother is killed. The audience does not mind because it is deftly handled and fuels the dramatic conflict of the film.
In "Stranger Than Fiction," the entire movie is based on the coincidences in the life of Harold Crick, skillfully managed by the author in the film, Karen Eiffel.
The audience balks at a film when coincidence is illogical, used to fill plot holes or weak character action.
Gretchen Gersh Whitman

There's also a whole subgenre of films - usually black comedies - that are built around coincidence as the main structural support beam. I love films like this, which usually have an ensemble cast of characters whose lives intersect and intertwine in the most lip-smackingly delicious, sometimes downright ridiculous, ways. Alfred Hitchcock made numerous movies where coincidence played such a major factor, you could say the film was about coincidence itself. This was also the stock and trade of the Seinfeld series. While the show's creators always said it was a show about nothing, structurally and thematically, it was a show about coincidence. Three or four separate story lines were put into play during the first act. By the final commercial break, those lines had always converged into one.

As far as Shawshank goes, there's another reason the Tommy coincidence works. Tommy was in prison with Red for a long, long time before he put two-and-two together. If he would have walked in, sat down to lunch with the fellas, and said something like, "Hey - aren't you that banker who was accused of murdering his wife? Boy - do I know something you're gonna wish you'd known all along!" we would have all been torn out of the movie, no matter how much we cared about Andy. Instead, Tommy had been in prison for a few years before he finally adds it all up, and when he does so, he's talking to Red, not Andy. This rang true for me because there's a psychological truth to it. This is how our minds work. Tommy got new information about Andy from Red, and that's what triggered his Ah-ha moment. So it feels real, not like the glaring hand of the writer.

Luzid, ggw07, and Annika W: Each of your three comments are terrific and spot-on. In fact, I'll update the OP with your insights.

In Casablanca, I don't see Laszlo's arrival at Rick's as a coincidence -- when Ugarte gives Rick the letters for safekeeping, he says outright that he's planning to sell the letters of transit that night. Laszlo arrives at Rick's stating he has a reservation and looking for Ugarte, clearly having arranged to meet him there. The murder and theft of the letters of transit seem more like a commissioned job, as Laszlo is especially in need of papers that cannot be questioned -- a normal exit visa is unavailable to him.

The script is also forthright in highlighting the inexorable forces that will bring everyone together at Rick's: first, in the opening narration, Casablanca is established as the last stop on the refugee trail before Lisbon and then the new world. Then, Renault tells Strasser that "everybody comes to Rick's".

To me, the crucial coincidence is that Laszlo is the husband Ilsa thought dead when she and Rick were together in Paris. When she discovers Laszlo is alive and leaves Rick waiting on the train platform with his insides kicked out, that turns him into the cynical saloonkeeper who sticks his neck out for nobody.

Their arrival in Casablanca and at Rick's provides the impetus for him to change, once again becoming the person hinted at in Strasser's biographical dossier the Paris flashbacks -- a "rank sentimentalist", as Renault puts it.

There are many reasons Casablanca can be described as a perfect movie, but the tight internal logic is a huge component, bringing all of the characters together so the story can "just happen" -- before today, I never really thought much about all of the "coincidences" because they just didn't matter.

@Ken King: Great point and odd that after having seen Casablanca several times, I didn't draw the connection between Ugarte's potential sale and Laszlo. But of course, that's true.

I agree about Casablanca. We are told that a "very special man" (or however Ugarte put it) is was slated to show up that night. Rick know. Strasser knows. Renault knows. It's like all of Cassablanca is waiting for the dirt to go down that night at Rick's. But no one knew that the woman on Victor Laszlo's arm was going to be Rick's old flame, and therein likes the coinky-dink.

I agree about Casablanca. We are told that a "very special man" (or however Ugarte put it) is was slated to show up that night. Rick know. Strasser knows. Renault knows. It's like all of Cassablanca is waiting for the dirt to go down that night at Rick's. But no one knew that the woman on Victor Laszlo's arm was going to be Rick's old flame, and therein likes the coinky-dink.