Really, if you evaluate Style["/", Underlined] or type / and set the font of the character to be underlined through Format|Font..., you get $/$, not $\underline{/}$.
What I want to do is to underline a hyperlink, but...
the output is ugly. Note, that line has a gap before ".com".
This is not the problem of 'Courier New' only. I think this behavior caused because /, //, ., etc. are operators.
Answer
You can use UnderBar instead of Style[...,Underlined]:
Hyperlink[UnderBar[#], #] &@ "http://www.wolfram.com/Sine.html"

Note: The issue you observe is mentioned in the docs Underlined >> Possible Issues:
Underlined will recursively affect all elements of an expression
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