If $X$ and $Y$ are independent random variables, then their Spearman's Correlation Coefficient is...? I have often seen the following statement:

If $X$ and $Y$ are independent random variables, then their 'correlation' is $0$

What is meant by 'correlation'? Is it specific to measures of linear correlation? Or does this claim apply to non-linear measures of correlation, as well?  (e.g. if $X$ and $Y$ are independent variables, then their Spearman's Correlation Coefficient is...?)

This typically means Pearson correlation. While zero Pearson correlation is insufficient to conclude that two variables are independent (think of a parabolic relationship like I show here), a nonzero correlation means dependence.
However, it applies to all types of correlation. If independence is to have a reasonable definition in probability theory (which I believe it does), then knowing something about one variable should tell you nothing about the other. For Spearman correlation in particular, if variables are independent, then knowing that one variable takes a high value should not tell you that the other is particularly likely to take a high value (or a low value for Spearman correlation less than zero).
