Solution:
To make a Roman numeral using matchsticks, we need to consider the number of straight lines required to form the Roman numeral.

The following are the Roman numerals and the number of straight lines required to form them using matchsticks:

- I - 1 straight line
- V - 2 straight lines
- X - 2 straight lines
- L - 3 straight lines
- C - 4 straight lines
- D - 5 straight lines
- M - 6 straight lines

To form a Roman numeral using 10 matchsticks, we can use the following numerals:

- IIIIIIIIII - 9 straight lines (equivalent to 9 in Roman numerals)
- VIIIIII - 8 straight lines (equivalent to 9 in Roman numerals)
- VVIIIII - 7 straight lines (equivalent to 7 in Roman numerals)
- XXXVII - 6 straight lines (equivalent to 37 in Roman numerals)
- XXXIII - 5 straight lines (equivalent to 33 in Roman numerals)
- XXVIIII - 4 straight lines (equivalent to 29 in Roman numerals)
- XXVIII - 4 straight lines (equivalent to 28 in Roman numerals)
- XXIIIIII - 3 straight lines (equivalent to 24 in Roman numerals)
- XXIII - 3 straight lines (equivalent to 23 in Roman numerals)
- XVIIII - 2 straight lines (equivalent to 19 in Roman numerals)
- XVIII - 2 straight lines (equivalent to 18 in Roman numerals)
- XIII - 2 straight lines (equivalent to 13 in Roman numerals)
- VII - 2 straight lines (equivalent to 7 in Roman numerals)

Therefore, the only number in the given options that can be represented using 10 matchsticks is 37 (equivalent to XXXVII in Roman numerals). Hence, the correct answer is option C.