The following script gets a list of the synonyms, parses them, and then the checks if the existence of the object on the sys.objects table.

Any errors encountered are assumed as "not a valid synonym"

/*
Author: Miguel Quintana

Description:
Reads all SYNONYMS on the database and test if they are valid.
It checks their EXISTENCE on the SYSOBJECTS table
and uses the Link Server if available

No configuration needed. Just run it under the DB that you are testing Synonyms

*/


-- this variables are for the cursor
DECLARE @NAME NVARCHAR(100)
  ,@SERVERNAME NVARCHAR(20)
  ,@DBName NVARCHAR(100)
  ,@SCHEMA NVARCHAR(100)
  ,@OBJECTNAME NVARCHAR(1000)
  ,@BaseObjectName NVARCHAR(1000)
-- This are to process the data
DECLARE @CMD NVARCHAR(MAX)
DECLARE @RESULTS TABLE (
  VALUE NVARCHAR(100)
  ,SYNONYM NVARCHAR(100)
  ,BaseObjectName NVARCHAR(100)
  ,Link_SERVER NVARCHAR(100)
  )
DECLARE CUR CURSOR
FOR
SELECT NAME
  ,PARSENAME(base_object_name, 4) ServerName
  ,COALESCE(PARSENAME(base_object_name, 3), DB_NAME(DB_ID())) AS dbName
  ,COALESCE(PARSENAME(base_object_name, 2), SCHEMA_NAME(SCHEMA_ID())) AS schemaName
  ,PARSENAME(base_object_name, 1) AS objectName
  ,BASE_OBJECT_NAME
FROM sys.synonyms
order by ServerName DESC, NAME



OPEN CUR
FETCH NEXT FROM CUR INTO 
   @name
  ,@SERVERNAME
  ,@DBNAME
  ,@Schema
  ,@ObjectName
  ,@BaseObjectName
WHILE @@FETCH_STATUS = 0
BEGIN
  -- Test SYNONYMS that use Link Servers
  IF @SERVERNAME IS NOT NULL
  BEGIN
    SET @cmd = 'EXEC (''IF EXISTS(select name from ' + @DBName + '.[sys].[objects] where name = ''''' + @OBJECTNAME + ''''')
    SELECT ''''VALID SYNONYM'''',''''' + @NAME + ''''',''''' + @BaseObjectName + ''''',''''' + @SERVERNAME + '''''
    ELSE
    SELECT ''''NOT A VALID SYNONYM'''',''''' + @NAME + ''''',''''' + @BaseObjectName + ''''',''''' + @SERVERNAME + ''''''') AT [' + @SERVERNAME + ']'
    print @cmd
	BEGIN TRY
      INSERT INTO @RESULTS
	  EXEC (@cmd)
    END TRY
    -- IF the exec fails for any reason, the SYNONYM is not valid
    BEGIN CATCH
      INSERT INTO @RESULTS
      VALUES (
        'NOT A VALID SYNONYM'
        ,@NAME
        ,@BaseObjectName
        ,@SERVERNAME
        )
    END CATCH
  END
  -- Test SYNONYMS that do not use Link Servers
  ELSE
  BEGIN
    SET @cmd = 'EXEC (''IF EXISTS(select name from ' + @DBName + '.[sys].[objects] where name = ''''' + @OBJECTNAME + ''''')
    SELECT ''''VALID SYNONYM'''',''''' + @NAME + ''''',''''' + @BaseObjectName + ''''','''' ''''
    ELSE
    SELECT ''''NOT A VALID SYNONYM'''',''''' + @NAME + ''''',''''' + @BaseObjectName + ''''',''''' + @@SERVERNAME + ''''''')'
    print @cmd    
	
	BEGIN TRY
      INSERT INTO @RESULTS
      EXEC (@cmd)
    END TRY
    -- IF the exec fails for any reason, the SYSNONYM is not valid
    BEGIN CATCH
      INSERT INTO @RESULTS
      VALUES (
        'NOT A VALID SYNONYM'
        ,@NAME
        ,@BaseObjectName
        ,@@SERVERNAME
        )
    END CATCH
  END
  FETCH NEXT FROM CUR INTO
     @name
    ,@SERVERNAME
    ,@DBNAME
    ,@Schema
    ,@ObjectName
    ,@BaseObjectName
END
CLOSE CUR
DEALLOCATE CUR

SELECT Value
  ,[SYNONYM]
  ,UPPER(Link_SERVER) 'Link Server'
  ,BaseObjectName
  ,COALESCE(ss.data_source,'') 'Data Source'
FROM @RESULTS r
LEFT JOIN master.sys.servers ss on r.Link_SERVER = ss.name


ORDER BY VALUE

Link Servers are supported.

Sometimes, it will report as not valid SYNONYM even though you are able to use them (select top 1 * from <sn_syonym_name>). but users without sysadmin cannot.

There are many reasons why that would happen, usually drop & recreate fixes the issue or even running the following:

EXEC sp_serveroption @server = '',@optname = 'remote proc transaction promotion', @optvalue = 'false' ;