I think that the doc comment "<param name="strTable">The table to retrieve schema information for, must not be null</param>" of SQLiteConnection.Schema_Columns(string strCatalog, string strTable, string strColumn) is not correct, as strTable *can* be null. 
The check happens here:
if (String.IsNullOrEmpty(strTable) || String.Compare(strTable, rdTables.GetString(2), StringComparison.OrdinalIgnoreCase) == 0)
{ 
    ....
So the method call mySQLiteConnection.GetSchema("Columns", new string[] { null, null, "MY_COLUMN" }) works like most other ADO.NET implementations and returns information about every "MY_COLUMN" column in the database.

Fixed on trunk via check-in [7632e932f50e36d8].