First, the subject line: there is no smallest rational square root as there is no largest integer square root. They would be reciprocal. Just remember that any natural number - any positive number, in fact - is the root of its square.

>It seems like only perfect squares
>and multiples of perfect squares (4*49) have obviously
>rational square roots

That's correct. A square root of a non-perfect square (integer or rational) is irrational.

>Is there a way to
>tell the difference between a true irrational number and a
>huge rational number?

By looking at the decimal expansion? The only way to claim that a decimal expansion is that of a rational number is to observe a period, i.e., a repeated pattern. This, even if short, may start arbtrarily far from the decimal point.

>(1/499 repeats for 500 digits)

It also may be arbitrarily long, as this example shows.

To understand this imagine a rational number and its decimal expansion which is periodic. Now, you start changing some digits randomly. If you change a random digit in the recurring period it will no longer will be recurring and the resulting number will be irrational.

> And
>finally, of all those irrational-looking decimal numbers,
>which is the first one (not a perfect square) to be
>deceptively rational?

I do not understand that question but suspect that the answer is there is no first in whatever sense.

>I have my money on 18^.5, but I do not know how to tell what
>is what.

Why? What do you see in 18^.5?