Is there a well-known secular sentence that uses all three of the imperative, indicative, and subjunctive moods?

Well, subjunctive mood is a bit opaque to me, and it's more than one sentence, but perhaps

Just sit right back

And you'll hear a tale

A tale of a fateful trip,

That started from this tropic port,

Aboard this tiny ship.

...

The weather started getting rough,

The tiny ship was tossed;

If not for the courage of the fearless crew

The Minnow would be lost.

(The Minnow would be lost.)

"Just sit right back" sounds like imperative to me, and I think "If not for the courage of the fearless crew, the Minnow would be lost" matches some of the examples I've found of subjunctive mood, with everything else in indicative.

"Star night, Star bright.
First Star I've seen tonight,
Wish I may, wish I might
Have the wish I wish to-night"

First Star is invoked to "wish" (imperative) something for me.
What that is is in the subjunctive, "that I may have, that I might have..."
"the wish I wish tonight" is in the indicative.

This anthology gives a different version which is parsed differently.

'Subjunctives' (meaning may clauses that require subjunctive in Latin) survive in a few granny quotes, such as "I would'n if I were you!"
and in a few incantations.