$16$ pieces of the shape $1\times 3$ are placed on a $7\times 7$ chessboard, each of which is exactly three fields. One field remains free. Find all possible positions of this field.
To solve this problem, we need to determine the possible positions of the single free field on a $7 \times 7$ chessboard when $16$ pieces of size $1 \times 3$ are placed on it. Each piece covers exactly three fields, leaving one field uncovered.

1. **Calculate the total number of fields:**
   \[
   7 \times 7 = 49 \text{ fields}
   \]
   Since each $1 \times 3$ piece covers 3 fields, the total number of fields covered by 16 pieces is:
   \[
   16 \times 3 = 48 \text{ fields}
   \]
   Therefore, one field remains uncovered:
   \[
   49 - 48 = 1 \text{ field}
   \]

2. **Color the chessboard in two different ways:**
   - **First Coloring:**
     Color the chessboard in a checkerboard pattern with colors $A$ and $B$ such that no two adjacent fields have the same color. This results in:
     \[
     \begin{array}{ccccccc}
     A & B & A & B & A & B & A \\
     B & A & B & A & B & A & B \\
     A & B & A & B & A & B & A \\
     B & A & B & A & B & A & B \\
     A & B & A & B & A & B & A \\
     B & A & B & A & B & A & B \\
     A & B & A & B & A & B & A \\
     \end{array}
     \]
   - **Second Coloring:**
     Color the chessboard in a different pattern with colors $C$ and $D$ such that no two adjacent fields have the same color. This results in:
     \[
     \begin{array}{ccccccc}
     C & D & C & D & C & D & C \\
     D & C & D & C & D & C & D \\
     C & D & C & D & C & D & C \\
     D & C & D & C & D & C & D \\
     C & D & C & D & C & D & C \\
     D & C & D & C & D & C & D \\
     C & D & C & D & C & D & C \\
     \end{array}
     \]

3. **Analyze the positions of the $1 \times 3$ pieces:**
   Each $1 \times 3$ piece covers three consecutive fields. In the first coloring, each piece will cover either two $A$ fields and one $B$ field or two $B$ fields and one $A$ field. Similarly, in the second coloring, each piece will cover either two $C$ fields and one $D$ field or two $D$ fields and one $C$ field.

4. **Determine the possible positions of the free field:**
   The free field must be a field that can be left uncovered by the $1 \times 3$ pieces in both colorings. This means the free field must be a field that is $A$ in the first coloring and $C$ in the second coloring. By examining the colorings, we see that the fields that are $A$ in the first coloring and $C$ in the second coloring are:
   \[
   \begin{array}{ccccccc}
   A & B & A & B & A & B & A \\
   B & A & B & A & B & A & B \\
   A & B & A & B & A & B & A \\
   B & A & B & A & B & A & B \\
   A & B & A & B & A & B & A \\
   B & A & B & A & B & A & B \\
   A & B & A & B & A & B & A \\
   \end{array}
   \]
   and
   \[
   \begin{array}{ccccccc}
   C & D & C & D & C & D & C \\
   D & C & D & C & D & C & D \\
   C & D & C & D & C & D & C \\
   D & C & D & C & D & C & D \\
   C & D & C & D & C & D & C \\
   D & C & D & C & D & C & D \\
   C & D & C & D & C & D & C \\
   \end{array}
   \]
   The fields that are $A$ in the first coloring and $C$ in the second coloring are:
   \[
   \begin{array}{ccccccc}
   A &   & A &   & A &   & A \\
     & A &   & A &   & A &   \\
   A &   & A &   & A &   & A \\
     & A &   & A &   & A &   \\
   A &   & A &   & A &   & A \\
     & A &   & A &   & A &   \\
   A &   & A &   & A &   & A \\
   \end{array}
   \]

5. **Conclude the possible positions of the free field:**
   The free field can be any of the fields that are $A$ in the first coloring and $C$ in the second coloring. These fields are:
   \[
   (1,1), (1,3), (1,5), (1,7), (3,1), (3,3), (3,5), (3,7), (5,1), (5,3), (5,5), (5,7), (7,1), (7,3), (7,5), (7,7)
   \]

The final answer is $\boxed{(1,1), (1,3), (1,5), (1,7), (3,1), (3,3), (3,5), (3,7), (5,1), (5,3), (5,5), (5,7), (7,1), (7,3), (7,5), (7,7)}$