Which of the following is the smallest value?

The smallest value among the given options is indeed 0.07. When comparing decimal numbers, it is important to look at the digits from left to right.

Starting with the first digit after the decimal point, 0.07 has '0' in the tenths place and '7' in the hundredths place. The next option, 0.12, also starts with '0' in the tenths place, but has '1' in the hundredths place, which is larger than 7.

Moving on to the other values: 3.21 has '3' in the whole number part, and 1.564 has '1' in the whole number part as well, both of which are greater than 0.07. Therefore, both of these options are also larger when considering the whole numbers.

In short, 0.07 is the smallest because it is less than both 0.12 and the whole number values of 1.564 and 3.21, which are significantly larger. This analysis confirms that 0.07 is indeed the smallest value among the choices provided.