Here is a solution for the single column search using PATINDEX.
It also displays the StartPosition, InvalidCharacter and ASCII code.

select line,
  patindex('%[^ !-~]%' COLLATE Latin1_General_BIN,Line) as [Position],
  substring(line,patindex('%[^ !-~]%' COLLATE Latin1_General_BIN,Line),1) as [InvalidCharacter],
  ascii(substring(line,patindex('%[^ !-~]%' COLLATE Latin1_General_BIN,Line),1)) as [ASCIICode]
from  staging.APARMRE1
where patindex('%[^ !-~]%' COLLATE Latin1_General_BIN,Line) >0

I've been running this bit of code with success

declare @UnicodeData table (
     data nvarchar(500)
)
insert into 
    @UnicodeData
values 
    (N'Horse�')
    ,(N'Dog')
    ,(N'Cat')
select
    data
from
    @UnicodeData 
where
    data collate LATIN1_GENERAL_BIN != cast(data as varchar(max))

Which works well for known columns.

For extra credit, I wrote this quick script to search all nvarchar columns in a given table for Unicode characters.

declare 
    @sql    varchar(max)    = ''
    ,@table sysname         = 'mytable' -- enter your table here
;with ColumnData as (
    select
        RowId               = row_number() over (order by c.COLUMN_NAME)
        ,c.COLUMN_NAME
        ,ColumnName         = '[' + c.COLUMN_NAME + ']'
        ,TableName          = '[' + c.TABLE_SCHEMA + '].[' + c.TABLE_NAME + ']' 
    from
        INFORMATION_SCHEMA.COLUMNS c
    where
        c.DATA_TYPE         = 'nvarchar'
        and c.TABLE_NAME    = @table
)
select
    @sql = @sql + 'select FieldName = ''' + c.ColumnName + ''',         InvalidCharacter = [' + c.COLUMN_NAME + ']  from ' + c.TableName + ' where ' + c.ColumnName + ' collate LATIN1_GENERAL_BIN != cast(' + c.ColumnName + ' as varchar(max)) '  +  case when c.RowId <> (select max(RowId) from ColumnData) then  ' union all ' else '' end + char(13)
from
    ColumnData c
-- check
-- print @sql
exec (@sql)

I'm not a fan of dynamic SQL but it does have its uses for exploratory queries like this.

try something like this:

DECLARE @YourTable table (PK int, col1 varchar(20), col2 varchar(20), col3 varchar(20));
INSERT @YourTable VALUES (1, 'ok','ok','ok');
INSERT @YourTable VALUES (2, 'BA'+char(182)+'D','ok','ok');
INSERT @YourTable VALUES (3, 'ok',char(182)+'BAD','ok');
INSERT @YourTable VALUES (4, 'ok','ok','B'+char(182)+'AD');
INSERT @YourTable VALUES (5, char(182)+'BAD','ok',char(182)+'BAD');
INSERT @YourTable VALUES (6, 'BAD'+char(182),'B'+char(182)+'AD','BAD'+char(182)+char(182)+char(182));
--if you have a Numbers table use that, other wise make one using a CTE
WITH AllNumbers AS
(   SELECT 1 AS Number
    UNION ALL
    SELECT Number+1
        FROM AllNumbers
        WHERE Number<1000
)
SELECT 
    pk, 'Col1' BadValueColumn, CONVERT(varchar(20),col1) AS BadValue --make the XYZ in convert(varchar(XYZ), ...) the largest value of col1, col2, col3
    FROM @YourTable           y
        INNER JOIN AllNumbers n ON n.Number <= LEN(y.col1)
    WHERE ASCII(SUBSTRING(y.col1, n.Number, 1))<32 OR ASCII(SUBSTRING(y.col1, n.Number, 1))>127
UNION
SELECT 
    pk, 'Col2' BadValueColumn, CONVERT(varchar(20),col2) AS BadValue --make the XYZ in convert(varchar(XYZ), ...) the largest value of col1, col2, col3
    FROM @YourTable           y
        INNER JOIN AllNumbers n ON n.Number <= LEN(y.col2)
    WHERE ASCII(SUBSTRING(y.col2, n.Number, 1))<32 OR ASCII(SUBSTRING(y.col2, n.Number, 1))>127
UNION
SELECT 
    pk, 'Col3' BadValueColumn, CONVERT(varchar(20),col3) AS BadValue --make the XYZ in convert(varchar(XYZ), ...) the largest value of col1, col2, col3
    FROM @YourTable           y
        INNER JOIN AllNumbers n ON n.Number <= LEN(y.col3)
    WHERE ASCII(SUBSTRING(y.col3, n.Number, 1))<32 OR ASCII(SUBSTRING(y.col3, n.Number, 1))>127
order by 1
OPTION (MAXRECURSION 1000);

OUTPUT:

pk          BadValueColumn BadValue
----------- -------------- --------------------
2           Col1           BA¶D
3           Col2           ¶BAD
4           Col3           B¶AD
5           Col1           ¶BAD
5           Col3           ¶BAD
6           Col1           BAD¶
6           Col2           B¶AD
6           Col3           BAD¶¶¶
(8 row(s) affected)

This script searches for non-ascii characters in one column. It generates a string of all valid characters, here code point 32 to 127. Then it searches for rows that don't match the list:

declare @str varchar(128);
declare @i int;
set @str = '';
set @i = 32;
while @i <= 127
    begin
    set @str = @str + '|' + char(@i);
    set @i = @i + 1;
    end;
select  col1
from    YourTable
where   col1 like '%[^' + @str + ']%' escape '|';

running the various solutions on some real world data - 12M rows varchar length ~30, around 9k dodgy rows, no full text index in play, the patIndex solution is the fastest, and it also selects the most rows.

(pre-ran km. to set the cache to a known state, ran the 3 processes, and finally ran km again - the last 2 runs of km gave times within 2 seconds)

patindex solution by Gerhard Weiss -- Runtime 0:38, returns 9144 rows

select dodgyColumn from myTable fcc
WHERE  patindex('%[^ !-~]%' COLLATE Latin1_General_BIN,dodgyColumn ) >0

the substring-numbers solution by MT. -- Runtime 1:16, returned 8996 rows

select dodgyColumn from myTable fcc
INNER JOIN dbo.Numbers32k dn ON dn.number<(len(fcc.dodgyColumn ))
WHERE ASCII(SUBSTRING(fcc.dodgyColumn , dn.Number, 1))<32 
    OR ASCII(SUBSTRING(fcc.dodgyColumn , dn.Number, 1))>127

udf solution by Deon Robertson -- Runtime 3:47, returns 7316 rows

select dodgyColumn 
from myTable 
where dbo.udf_test_ContainsNonASCIIChars(dodgyColumn , 1) = 1

There is a user defined function available on the web 'Parse Alphanumeric'. Google UDF parse alphanumeric and you should find the code for it. This user defined function removes all characters that doesn't fit between 0-9, a-z, and A-Z.

Select * from Staging.APARMRE1 ar
where udf_parsealpha(ar.last_name) <> ar.last_name

That should bring back any records that have a last_name with invalid chars for you...though your bonus points question is a bit more of a challenge, but I think a case statement could handle it. This is a bit psuedo code, I'm not entirely sure if it'd work.

Select id, case when udf_parsealpha(ar.last_name) <> ar.last_name then 'last name'
when udf_parsealpha(ar.first_name) <> ar.first_name then 'first name'
when udf_parsealpha(ar.Address1) <> ar.last_name then 'Address1'
end, 
case when udf_parsealpha(ar.last_name) <> ar.last_name then ar.last_name
when udf_parsealpha(ar.first_name) <> ar.first_name then ar.first_name
when udf_parsealpha(ar.Address1) <> ar.last_name then ar.Address1
end
from Staging.APARMRE1 ar
where udf_parsealpha(ar.last_name) <> ar.last_name or
udf_parsealpha(ar.first_name) <> ar.first_name or
udf_parsealpha(ar.Address1) <> ar.last_name

I wrote this in the forum post box...so I'm not quite sure if that'll function as is, but it should be close. I'm not quite sure how it will behave if a single record has two fields with invalid chars either.

As an alternative, you should be able to change the from clause away from a single table and into a subquery that looks something like:

select id,fieldname,value from (
Select id,'last_name' as 'fieldname', last_name as 'value'
from Staging.APARMRE1 ar
Union
Select id,'first_name' as 'fieldname', first_name as 'value'
from Staging.APARMRE1 ar
---(and repeat unions for each field)
)
where udf_parsealpha(value) <> value

Benefit here is for every column you'll only need to extend the union statement here, while you need to put that comparisson three times for every column in the case statement version of this script

To find which field has invalid characters:

SELECT * FROM Staging.APARMRE1 FOR XML AUTO, TYPE

You can test it with this query:

SELECT top 1 'char 31: '+char(31)+' (hex 0x1F)' field
from sysobjects
FOR XML AUTO, TYPE

The result will be:

Msg 6841, Level 16, State 1, Line 3 FOR XML could not serialize the data for node 'field' because it contains a character (0x001F) which is not allowed in XML. To retrieve this data using FOR XML, convert it to binary, varbinary or image data type and use the BINARY BASE64 directive.

It is very useful when you write xml files and get error of invalid characters when validate it.

Here is a UDF I built to detectc columns with extended ascii charaters. It is quick and you can extended the character set you want to check. The second parameter allows you to switch between checking anything outside the standard character set or allowing an extended set:

create function [dbo].[udf_ContainsNonASCIIChars]
(
@string nvarchar(4000),
@checkExtendedCharset bit
)
returns bit
as
begin
    declare @pos int = 0;
    declare @char varchar(1);
    declare @return bit = 0;
    while @pos < len(@string)
    begin
        select @char = substring(@string, @pos, 1)
        if ascii(@char) < 32 or ascii(@char) > 126 
            begin
                if @checkExtendedCharset = 1
                    begin
                        if ascii(@char) not in (9,124,130,138,142,146,150,154,158,160,170,176,180,181,183,184,185,186,192,193,194,195,196,197,199,200,201,202,203,204,205,206,207,209,210,211,212,213,214,216,217,218,219,220,221,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,248,249,250,251,252,253,254,255)
                            begin
                                select @return = 1;
                                select @pos = (len(@string) + 1)
                            end
                        else
                            begin
                                select @pos = @pos + 1
                            end
                    end
                else
                    begin
                        select @return = 1;
                        select @pos = (len(@string) + 1)    
                    end
            end
        else
            begin
                select @pos = @pos + 1
            end
    end
    return @return;
end

USAGE:

select Address1 
from PropertyFile_English
where udf_ContainsNonASCIIChars(Address1, 1) = 1

I took Gerhard Weiss's script and made it my own as my business scenario was pretty simple and only needed to update 10K rows.

--CONTAINS() function requires a full text catalog
--interesting to me is that CONTAINS() can find the ASCII characters, but REPLACE() cannot
WITH cte_AsciiCharacterList AS
(select LineID
    , LineSyntax
    , Position = patindex('%[^ !-~]%' COLLATE Latin1_General_BIN,LineSyntax)
    , InvalidCharacter = substring(LineSyntax,patindex('%[^ !-~]%' COLLATE Latin1_General_BIN,LineSyntax),1)
    , ASCIICode = ascii(substring(LineSyntax,patindex('%[^ !-~]%' COLLATE Latin1_General_BIN,LineSyntax),1))
    , LineSyntaxPart1 = SUBSTRING(LineSyntax, 1, patindex('%[^ !-~]%' COLLATE Latin1_General_BIN,LineSyntax) - 1)
    , LineSyntaxPart2 = SUBSTRING(LineSyntax, patindex('%[^ !-~]%' COLLATE Latin1_General_BIN,LineSyntax) + 3, 10000)
    , NewLineSyntax =  SUBSTRING(LineSyntax, 1, patindex('%[^ !-~]%' COLLATE Latin1_General_BIN,LineSyntax) - 1) + '-' + SUBSTRING(LineSyntax, patindex('%[^ !-~]%' COLLATE Latin1_General_BIN,LineSyntax) + 3, 10000)
FROM stg.TableName
WHERE Contains(linesyntax, '"ΓÇö"')
)
UPDATE stg.TableName
SET LineSyntax = NewLineSyntax
FROM stg.TableName x
INNER JOIN cte_AsciiCharacterList y on x.LineID = y.LineID