What is the smallest possible value of $\left|12^m-5^n\right|$, where $m$ and $n$ are positive integers?
To find the smallest possible value of $\left|12^m - 5^n\right|$ where $m$ and $n$ are positive integers, we start by evaluating some initial values and then proceed to prove that the smallest possible value is indeed $7$.

1. **Initial Evaluation:**
   \[
   \left|12^1 - 5^1\right| = \left|12 - 5\right| = 7
   \]
   This gives us a candidate for the smallest value. We now need to prove that no smaller value is possible.

2. **Proof by Contradiction:**
   Suppose there exists a nonnegative integer $k < 7$ that satisfies $\left|12^m - 5^n\right| = k$. Since $12^m$ is even and $5^n$ is odd, the difference between them must be odd. Thus, $k$ must be $1$, $3$, or $5$.

3. **Eliminating $k = 3$ and $k = 5$:**
   - For $k = 3$: 
     \[
     3 \nmid 5^n \quad \text{and} \quad 3 \nmid 12^m
     \]
     Hence, $k = 3$ is not possible.
   - For $k = 5$: 
     \[
     5 \nmid 12^m
     \]
     Hence, $k = 5$ is not possible.

4. **Case $k = 1$:**
   We must have:
   \[
   12^m - 5^n = \pm 1 \implies 5^n = 12^m \pm 1
   \]

   - **Case 1: $5^n = 12^m - 1$**
     \[
     12^m - 1 \equiv 0 \pmod{11}
     \]
     Since $12 \equiv 1 \pmod{11}$, we have:
     \[
     1^m - 1 \equiv 0 \pmod{11} \implies 0 \equiv 0 \pmod{11}
     \]
     This is always true, but it does not provide any new information. We need to check if $5^n$ can be $12^m - 1$ for any $m$ and $n$.

   - **Case 2: $5^n = 12^m + 1$**
     \[
     12^m + 1 \equiv 0 \pmod{5}
     \]
     Since $12 \equiv 2 \pmod{5}$, we have:
     \[
     2^m + 1 \equiv 0 \pmod{5} \implies 2^m \equiv -1 \pmod{5}
     \]
     The powers of $2$ modulo $5$ cycle as follows: $2, 4, 3, 1$. Therefore, $2^m \equiv -1 \pmod{5}$ implies:
     \[
     m \equiv 2 \pmod{4}
     \]
     Let $m = 4p + 2$ for some integer $p$. Then:
     \[
     12^{4p + 2} + 1 = 5^n \implies (12^2)^{2p + 1} + 1 = 5^n \implies 144^{2p + 1} + 1 = 5^n
     \]
     By the sum of nth powers factorization, the left-hand side is divisible by $145$, meaning it is divisible by $29$. However, $5^n$ is not divisible by $29$, leading to a contradiction.

Since all cases for $k < 7$ lead to contradictions, the smallest possible value of $\left|12^m - 5^n\right|$ is indeed $7$.

The final answer is $\boxed{7}$.