This phrase is more common in UK English than other varieties, so it's especially useful when talking to Brits. Many's the time I've heard this phrase on my trips to London.
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Reflective Tone
'Many’s the…' is often used when reflecting on personal experiences or memories. Many's the night I've sat up contemplating the stars.
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Emotional Emphasis
The phrase can carry an emotional weight, suggesting frequency mixed with nostalgia or sentiment. Many's the day I've missed my homeland.