Arrange the following fractions from the smallest to the largest. $\frac{3}{2}, \quad 1\frac{5}{9}, \quad \frac{6}{5}$

1. Convert all fractions to mixed numbers or decimals to compare their values

$\frac{3}{2} = 1\frac{1}{2} = 1.5$
$1\frac{5}{9} \approx 1.555...$
$\frac{6}{5} = 1\frac{1}{5} = 1.2$

2. Arrange the values from smallest to largest

Comparing the values: $1.2 < 1.5 < 1.555...$
Therefore, the order is $\frac{6}{5}, \frac{3}{2}, 1\frac{5}{9}$.
This matches option (1).