To solve the expression \( 12 \times [ 390 \div [ 390 \div [500 - 37 \times 10]] \), we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

1. **Evaluate the innermost expression first:**
\[
500 - 37 \times 10
\]
Calculate the multiplication inside the brackets:
\[
37 \times 10 = 370
\]
Then subtract from 500:
\[
500 - 370 = 130
\]

2. **Substitute back into the expression:**
\[
12 \times [ 390 \div [ 390 \div 130]]
\]

3. **Evaluate the next innermost expression:**
\[
390 \div 130
\]
Perform the division:
\[
390 \div 130 = 3
\]

4. **Substitute back into the expression:**
\[
12 \times [ 390 \div 3]
\]

5. **Evaluate the next division:**
\[
390 \div 3
\]
Perform the division:
\[
390 \div 3 = 130
\]

6. **Substitute back into the expression:**
\[
12 \times 130
\]

7. **Finally, perform the multiplication:**
\[
12 \times 130 = 1560
\]

So, the value of the expression \( 12 \times [ 390 \div [ 390 \div [500 - 37 \times 10]] ] \) is \( 1560 \).