The best way to explain what syllepses are is really by example. (All examples stolen from Wikipedia unless mentioned otherwise.)
He took my advice and my wallet (The Free Dictionary)
On his fishing trip, he caught three trout and a cold. (Dictionary.com)
If we don't hang together, we shall hang separately (Benjamin Franklin)Oh, flowers are as common here, Miss Fairfax, as people are in London. (The Importance of Being Earnest)
Are you getting fit or having one? (M*A*S*H)
You are free to execute your laws, and your citizens, as you see fit. (Star Trek: The Next Generation)
He leaned heavily on the lectern and stale jokes.
He said, as he hastened to put out the cat, the wine, his cigar and the lamps... ("Madeira M'Dear")
You can leave in a taxi. If you can't get a taxi, you can leave in a huff. If that's too soon, you can leave in a minute and a huff. (Groucho Marx)
She stirred my soul and my risotto.
You can find a few more here.
Interestingly, a sentence is also a zeugma or syllepsis if part of the sentence is applied improperly to the wrong half of the sentence. For example:
To wage war and peace (Dictionary.com)
A zeugma, more broadly, can be any phrase joined by a common noun or verb.
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