SQL: how to get all the distinct characters in a column, across all rows

Use this (shall work on any CTE-capable RDBMS):

select x.v into prod from (values('product1'),('widget2'),('nicknack3')) as x(v);

Test Query:

with a as 
(
    select v, '' as x, 0 as n from prod 
    union all
    select v, substring(v,n+1,1) as x, n+1 as n from a where n < len(v)
)
select v, x, n from a -- where n > 0
order by v, n
option (maxrecursion 0)

Final Query:

with a as 
(
    select v, '' as x, 0 as n from prod 
    union all
    select v, substring(v,n+1,1) as x, n+1 as n from a where n < len(v)
)
select distinct x from a where n > 0
order by x
option (maxrecursion 0)

Oracle version:

with a(v,x,n) as 
(
    select v, '' as x, 0 as n from prod 
    union all
    select v, substr(v,n+1,1) as x, n+1 as n from a where n < length(v)
)
select distinct x from a where n > 0

Here's a query that returns each character as a separate row, along with the number of occurrences. Assuming your table is called 'Products'

WITH ProductChars(aChar, remain) AS (
   SELECT LEFT(productName,1), RIGHT(productName, LEN(productName)-1) 
      FROM Products WHERE LEN(productName)>0
   UNION ALL
   SELECT LEFT(remain,1), RIGHT(remain, LEN(remain)-1) FROM ProductChars
      WHERE LEN(remain)>0
)
SELECT aChar, COUNT(*) FROM ProductChars
GROUP BY aChar

To combine them all to a single row, (as stated in the question), change the final SELECT to

SELECT aChar AS [text()] FROM
  (SELECT DISTINCT aChar FROM ProductChars) base
FOR XML PATH('')

The above uses a nice hack I found here, which emulates the GROUP_CONCAT from MySQL.

The first level of recursion is unrolled so that the query doesn't return empty strings in the output.

Given that your column is varchar, it means it can only store characters from codes 0 to 255, on whatever code page you have. If you only use the 32-128 ASCII code range, then you can simply see if you have any of the characters 32-128, one by one. The following query does that, looking in sys.objects.name:

with cteDigits as (
    select 0 as Number
    union all select 1 as Number
    union all select 2 as Number
    union all select 3 as Number
    union all select 4 as Number
    union all select 5 as Number
    union all select 6 as Number
    union all select 7 as Number
    union all select 8 as Number
    union all select 9 as Number)
, cteNumbers as (
    select U.Number + T.Number*10 + H.Number*100 as Number
    from cteDigits U
    cross join cteDigits T
    cross join cteDigits H)
, cteChars as (
    select CHAR(Number) as Char
    from cteNumbers 
    where Number between 32 and 128)
select cteChars.Char as [*]
from cteChars
cross apply (
    select top(1) *
    from sys.objects
    where CHARINDEX(cteChars.Char, name, 0) > 0) as o
for xml path('');