If $a$, $b$ and $c$ are the lengths of the sides of a triangle, all of them are not zero, it follows that $a, b, c \in \{1, 2, \dots, 9\}$.

i. If the triangle we construct is equilateral, let $n_1$ be the number of such 3-digit numbers. Since the three digits in such 3-digit number are equal, we have
$$n_1 = C_9^1 = 9.$$

ii. If the triangle we construct is isosceles but not equilateral, let $n_2$ be the number of such 3-digit numbers. Since there are only 2 different digits in such a 3-digit number, denote them by $a$ and $b$. Note that the equal sides and the base of an isosceles triangle can be replaced by each other, thus the number of such pairs $(a, b)$ is $2C_3^2$. But if the bigger number, say $a$, is the length of the base, then $a$ must satisfy the condition $b < a < 2b$. All pairs that do not satisfy this condition we list in the following table. There are 20 pairs.