Better Techniques for Trimming Leading Zeros in Sql Server?

Why don't you just cast the value to INTEGER and then back to VARCHAR?

SELECT  CAST(CAST('000000000' AS INTEGER) AS VARCHAR)

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       0

Other answers here to not take into consideration if you have all-zero's (or even a single zero).
Some always default an empty string to zero, which is wrong when it is supposed to remain blank.
Re-read the original question. This answers what the Questioner wants.

Solution #1:

--This example uses both Leading and Trailing zero's.
--Avoid losing those Trailing zero's and converting embedded spaces into more zeros.
--I added a non-whitespace character ("_") to retain trailing zero's after calling Replace().
--Simply remove the RTrim() function call if you want to preserve trailing spaces.
--If you treat zero's and empty-strings as the same thing for your application,
--  then you may skip the Case-Statement entirely and just use CN.CleanNumber .
DECLARE @WackadooNumber VarChar(50) = ' 0 0123ABC D0 '--'000'--
SELECT WN.WackadooNumber, CN.CleanNumber,
       (CASE WHEN WN.WackadooNumber LIKE '%0%' AND CN.CleanNumber = '' THEN '0' ELSE CN.CleanNumber END)[AllowZero]
 FROM (SELECT @WackadooNumber[WackadooNumber]) AS WN
 OUTER APPLY (SELECT RTRIM(RIGHT(WN.WackadooNumber, LEN(LTRIM(REPLACE(WN.WackadooNumber + '_', '0', ' '))) - 1))[CleanNumber]) AS CN
--Result: "123ABC D0"