Sherlock and Mycroft are playing Battleship on a $4\times4$ grid. Mycroft hides a single $3\times1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser?
To determine the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser, we need to consider the possible placements of the $3 \times 1$ cruiser on the $4 \times 4$ grid.

1. **Possible Placements of the Cruiser**:
   - The cruiser can be placed horizontally or vertically.
   - For horizontal placement, the cruiser can be placed in any of the 4 rows and can start in any of the first 2 columns (since it occupies 3 consecutive columns).
   - For vertical placement, the cruiser can be placed in any of the 4 columns and can start in any of the first 2 rows (since it occupies 3 consecutive rows).

2. **Total Number of Possible Placements**:
   - Horizontal placements: $4 \text{ rows} \times 2 \text{ starting columns} = 8$ placements.
   - Vertical placements: $4 \text{ columns} \times 2 \text{ starting rows} = 8$ placements.
   - Total placements: $8 + 8 = 16$ possible placements.

3. **Strategy to Guarantee a Hit**:
   - We need to ensure that every possible placement of the cruiser is hit by at least one of Sherlock's shots.
   - Consider the grid and the possible placements. If we fire at all the squares in the middle two rows and the middle two columns, we can cover all possible placements of the cruiser.

4. **Minimum Number of Shots**:
   - If we fire at the 4 central squares of the grid (positions (2,2), (2,3), (3,2), and (3,3)), we can ensure that at least one of these shots will hit the cruiser regardless of its placement.
   - However, we need to check if fewer than 4 shots can guarantee a hit.

5. **Verification**:
   - Suppose we try with 3 shots. We need to ensure that no matter where the cruiser is placed, at least one of these 3 shots will hit it.
   - If we place 3 shots in such a way that they do not cover all possible placements, there will be at least one placement of the cruiser that avoids all 3 shots.
   - Therefore, 3 shots are not sufficient to guarantee a hit.

6. **Conclusion**:
   - Since 3 shots are not sufficient, we need at least 4 shots.
   - By placing 4 shots in the central 4 squares, we can guarantee a hit.

The final answer is $\boxed{4}$.