Where's the 0xBEEF?

vim, 126 80 77 76

/\v1011\_.{9}(1110\_.{9}){2}1111<cr>:exe'norm Go'.join(getpos('.'))<cr>xxdawhPXXd{

Expects input in the form

111001111110
110100100000
010001111101
100100100100
100101100111
111111000010
110111000001
100111100001
100111011111
111110011111
100001010111
110011000011

And outputs (with 1-based indices) as

4 5

/                      regex search for...
\v                     enable "very magic" mode (less escaping)
1011\_.{9}             find the first line ("B"), followed by 8 chars + 1 \n
(1110\_.{9}){2}        find the second and third lines ("EE")
1111<cr>               find the fourth line ("F")
:exe'norm Go'.         insert at the beginning of the file...
join(getpos('.'))<cr>  the current position of the cursor
xxdawhPXX              do some finagling to put the numbers in the right order
d{                     delete the input

Thanks to Jörg Hülsermann for indirectly saving 46 bytes by making me realize my regex was super dumb, and to DJMcMayhem for 3 more bytes.

Jelly, 20 17 16 bytes

ṡ€4ḄZw€“¿ÇÇБĖUṀ

Input is in form of a Boolean matrix, output is the 1-based index pair (Y, X).

Try it online! or verify all test cases.

How it works

ṡ€4ḄZw€“¿ÇÇБĖUṀ  Main link. Argument: M (2D array of Booleans)

ṡ€4               Split each row into 9 chunks of length 4.
   Ḅ              Convert these chunks from base 2 to integer.
    Z             Zip/transpose. This places the columns of generated integers
                  into the rows of the matrix to comb through them.
       “¿ÇÇБ     Push the array of code points (in Jelly's code page) of these
                  characters, i.e., 0xB, 0xE, 0xE, and 0xF.
     w€           Window-index each; in each row, find the index of the contiguous
                  subarray [0xB, 0xE, 0xE, 0xF] (0 if not found).
                  Since the matrix contains on one BEEF, this will yield an array
                  of zeroes, with a single non-zero Y at index X.
             Ė    Enumerate; prefix each integer with its index.
              U   Upend; reverse the pairs to brings the zeroes to the beginning.
               Ṁ  Take the maximum. This yields the only element with positive
                  first coordinate, i.e., the pair [Y, X].

JavaScript (ES6), 63 60 56 bytes

s=>[(i=s.search(/1011.{9}(1110.{9}){2}1111/))%13,i/13|0]

Takes input as a 155-character space-delimited string of 12 12-digit binary strings, returns zero-indexed values. Edit: Saved 3 bytes thanks to @JörgHülsermann. Saved 4 bytes thanks to @ETHproductions.