Question: How can I determine that a particular file (which may or may not have an ".mp3" file extension) is in fact an MP3 file? I wish to do this in C#.

Answer: 
1. using file extension is not reliable.
2. the best libary you can use is <https://github.com/mono/taglib-sharp> it detects most of the common file types. maybe you just want mp3, so you can extract any mp3 related class.
3. a simpler library you can use is <https://github.com/judwhite/IdSharp>
Files often start with a "magic number" to identify the data format. Depending on the format, a file starts with a certain sequence of bytes which is unique to that format. There's no standard to follow so it's not 100% reliable. 

As fvu says, the mp3 magic number is `0x49 0x44 0x33`