How to get sp_executesql result into a variable?

Return values are generally not used to "return" a result but to return success (0) or an error number (1-65K). The above all seem to indicate that sp_executesql does not return a value, which is not correct. sp_executesql will return 0 for success and any other number for failure.

In the below, @i will return 2727

DECLARE @s NVARCHAR(500)
DECLARE @i INT;
SET @s = 'USE [Blah]; UPDATE STATISTICS [dbo].[TableName] [NonExistantStatisticsName];';
EXEC @i = sys.sp_executesql @s
SELECT @i AS 'Blah'

SSMS will show this Msg 2727, Level 11, State 1, Line 1 Cannot find index 'NonExistantStaticsName'.

DECLARE @tab AS TABLE (col1 VARCHAR(10), col2 varchar(10)) 
INSERT into @tab EXECUTE  sp_executesql N'
SELECT 1 AS col1, 2 AS col2
UNION ALL
SELECT 1 AS col1, 2 AS col2
UNION ALL
SELECT 1 AS col1, 2 AS col2'

SELECT * FROM @tab

DECLARE @vi INT
DECLARE @vQuery NVARCHAR(1000)

SET @vQuery = N'SELECT @vi= COUNT(*) FROM <TableName>'

EXEC SP_EXECUTESQL 
        @Query  = @vQuery
      , @Params = N'@vi INT OUTPUT'
      , @vi = @vi OUTPUT

SELECT @vi

If you have OUTPUT parameters you can do

DECLARE @retval int   
DECLARE @sSQL nvarchar(500);
DECLARE @ParmDefinition nvarchar(500);

DECLARE @tablename nvarchar(50)  
SELECT @tablename = N'products'  

SELECT @sSQL = N'SELECT @retvalOUT = MAX(ID) FROM ' + @tablename;  
SET @ParmDefinition = N'@retvalOUT int OUTPUT';

EXEC sp_executesql @sSQL, @ParmDefinition, @retvalOUT=@retval OUTPUT;

SELECT @retval;

But if you don't, and can not modify the SP:

-- Assuming that your SP return 1 value
create table #temptable (ID int null)
insert into #temptable exec mysp 'Value1', 'Value2'
select * from #temptable

Not pretty, but works.