The answer is all pairs such that $x,y$ have opposite signs. First of all, note that the condition clearly only depends on the value of $\frac{x}{y}$ so that we may w.l.o.g. assume that $x,y$ are coprime integers. First suppose that $x,y$ are of the same sign. Then we can find $a,b$ positive integers such that $bx-ay=1$. We claim that it is not possible to have $\frac{\{rx\}}{\{ry\}}=\frac{a}{b}$. Indeed, this would amount to $b\{rx\}=a\{ry\}$ and hence in particular to $r=brx-ary \in \mathbb{Z}$, which is clearly nonsense! On the other hand, assume that $x,y$ are of opposite signs. By changing to reciprocals, w.l.o.g. $x$ is positive and $y=-z$ is negative. Then clearly there are fractions $\frac{a}{x}<\frac{b}{z}$ such that between the two there are no multiples of $\frac{1}{x}$ or $\frac{1}{z}$. But then for $\frac{a}{x}<r<\frac{b}{z}$, our epression becomes $\frac{rx-a}{b-rz}$ which is easily seen to run through all positive rationals as $r$ runs through the specified interval. Done!