All the examples that I see for proving that two distributions are not computationally indistinguishable involve a pattern: choose a Distinguisher D(.) such that D(x) is 1 if x satisfies some certain properties.

For example, consider the distribution $X$ such that it contains all the binary strings in $\{0,1\}^k$ such that they end with 1 and distribution $Y$ uniformly spread over all strings in $\{0,1\}^k$. Now I can choose my distinguisher to be $D(x) = 1$, if $x$ ends with 1.

However, if there is no such pattern in our distributions, how can we prove that the given distributions are (not) computationally indistinguishable?

All the examples that I see for proving that two distributions are not computationally indistinguishable involve a pattern: choose a Distinguisher $D(\cdot)$ such that $D(x)$ is $1$ if $x$ satisfies some certain properties.

For example, consider the distribution $X$ such that it contains all the binary strings in $\{0,1\}^k$ such that they end with 1 and distribution $Y$ uniformly spread over all strings in $\{0,1\}^k$. Now I can choose my distinguisher to be $D(x) = 1$, if $x$ ends with $1$.

However, if there is no such pattern in our distributions, how can we prove that the given distributions are (not) computationally indistinguishable?