TextWriterTraceListener class
Possible issues

1 Follow a style guideline. It is common convention to camel case local variable names. 

var theFile = new FileInfo(...)  instead of  var TheFile = new FileInfo(...). 

Format your file equally (i.e. either all methods are separated by a blank line or none).

2 Keep lines short and use more variables to get better readability and easier debugging:

var newPath = Path.Combine(file.DirectoryName, file.Name, file.LastWriteTime.ToString(), file.Extension)
file.MoveTo(newPath);


instead of

TheFileInfo.MoveTo(string.Format(@"{0}\{1}_{2}{}", TheFileInfo.DirectoryName, TheFileInfo.Name, TheFileInfo.LastWriteTime.ToString(), TheFileInfo.Extension));


3 Use meaningful names. 

RollingTextWriterTraceListener instead of MyTextWriterTraceListener

[MethodImpl(MethodImplOptions.NoInlining)] instead of [System[...]NoInlining)]

What do you do if you need an other implementation - MyOtherTextWriterTraceListener?
The same applies to TheFileInfo, ConsoleApplication1.

4 Remove redundant qualifiers to remove clutter.

public MyTextWriterTraceListener(Stream stream) : base(stream) { }


instead of

public MyTextWriterTraceListener(System.IO.Stream stream) : base(stream) { }


var file = new FileInfo(...) instead FileInfo file = new FileInfo(...)

5 Are you sure you need the [MethodImpl(MethodImplOptions.NoInlining)] attribute?

What I like

1 if (TheFileInfo.Exists == false) return;

because the return statement reduces nesting (see arrow head anti pattern) and == false because its more obvious and than !TheFileInfo.Exists. A lonely ! can quick be overseen. I would recommend the same procedure for the following if statement, too:

if (TheFileInfo.LastWriteTime.Date &gt;= DateTime.Today) return;


instead of

if (TheFileInfo.LastWriteTime.Date &lt; DateTime.Today) { ... }


General recommendations

1 As k3b mentioned I would recommend using log4net, too. It has more flexibility you will ever need and you can configure it via config file which means you can change your logging output without recompiling.