Is there a sortable ('order by') alternative to CURRENT_TIMESTAMP
I have no idea why you would want to ORDER BY CURRENT_TIMESTAMP. If the table has a column where you stored CURRENT_TIMESTAMP when the row was created, you just need to say:

ORDER BY that_column_name;


Or if you want newest first:

ORDER BY that_column_name DESC;


... in other words, forget how the column was populated.

Now, if the comments are accurate, and you're actually not concerned so much about ordering but rather about filtering, then perhaps what you want is:

-- create a DATE variable based on today:

DECLARE @s DATE = CURRENT_TIMESTAMP;

-- convert it to the first of the month
-- (you can do this in the step above,
-- just separating it for clarity):

SET @s = DATEADD(DAY, 1-DAY(@s), @s);

-- use &gt;= and &lt; in a WHERE clause, and order by that column
-- (however this is just a coincidence, order by and filtering
-- are completely unrelated in this case:

SELECT [?] FROM dbo.[?]
  WHERE [datetime_column] &gt;= @s
  AND [datetime_column] &lt; DATEADD(MONTH, 1, @s)
  ORDER BY [datetime_column];


Here is why you don't want to use BETWEEN, in case anyone sends you that way:


https://sqlblog.org/2011/10/19/what-do-between-and-the-devil-have-in-common


Here is why you want to use built-in date functions and a native DATE type and not convert to a string to trim time from a datetime variable:


http://www.sqlperformance.com/2012/09/t-sql-queries/what-is-the-most-efficient-way-to-trim-time-from-datetime
http://www.sqlperformance.com/2012/10/t-sql-queries/trim-time


And here is some general information about bad practices in date/range queries:


https://sqlblog.org/2009/10/16/bad-habits-to-kick-mis-handling-date-range-queries
https://www.sentryone.com/blog/aaronbertrand/backtobasics-dating-responsibly
https://www.mssqltips.com/sqlservertip/5206/sql-server-datetime-best-practices/