Undecidability of the language of all Turing Machines with repeat strings as their language
Imagine a fixed machine $M$ and a fixed string $w$ and then let $M'$ be the machine that on input $x$ simulates $M$ on $w$, and if that simulation halts then answers Yes if $x=ab$ and No otherwise.
How could you decide whether $M'$ belongs to your language? You would need to know whether $M$ halts on $w$! As in that case $L(M')=\{ab\}$ and otherwise $L(M')= \varnothing$.
Details up to you :)