SqlConnection con = new SqlConnection(@"Data Source=samlab\sqlexpress;Initial Catalog=BillingCounters;Integrated Security=True;Pooling=False");
            try
            {
                con.Open();
                eventLog1.WriteEntry(con.State.ToString());
            }
            finally
            {
                con.Close();
            }

try
            {
                using (SqlConnection con = new SqlConnection(
                    @"Data Source=samlab\sqlexpress;Initial Catalog=BillingCounters;Integrated Security=True;Pooling=False"))
                { con.Open(); /* пишем в лог */ }
            }
            catch (Exception e) { /*пишем в лог e.Message*/ }

try
{
    using (SqlConnection con = new SqlConnection(s))
    {
        con.Open(); 
        /* пишем в лог */ 
    }
}
catch (Exception e) 
{ 
    /*пишем в лог e.Message*/ 
}

"Data Source=.\SQLEXPRESS;Initial Catalog=BillingCounters;Integrated Security=SSPI;"