Perl 6, 17 bytes

{[>](@_)+[>=] @_}

• Monotone strictly decreasing: 2
• Monotone non-increasing: 1
• Other: 0

Expanded:

{            # bare block lambda with implicit parameter list ｢@_｣
  [>]( @_ )  # reduce using ｢&infix:« > »｣
  +
  [>=] @_    # reduce using ｢&infix:« >= »｣
}

MATL, 10, 7 bytes

0hdX>ZS

Try it online! or verify all test cases!

3 bytes saved, thanks to @LuisMendo!

The outputs are

• Strictly decreasing: -1

• Non-increasing: 0

• Other: 1

Explanation:

0           % Push a '0'
 h          % Join the 0 to the end of the input array.
  d         % Get consecutive differences
   X>       % Get the largest difference
     ZS     % Get its sign
            % Implicitly print it

Mathematica, 22 bytes

Sign@*Max@*Differences

Unnamed function taking a list of numbers as input. Returns -1 if the list is strictly decreasing, 0 if it's nonincreasing but not strictly decreasing, and 1 if it's neither.

Pretty simple algorithm: take the differences of consecutive pairs, take the largest one, and take the sign of that largest one.

(I feel like there must exist some language in which this algorithm is 3 bytes....)

Regarding an array with a single entry: Differences yields an empty list; Max of an empty list gives -∞ (!); and Sign[-∞] evaluates to -1 (!!). So it actually works on this corner case. Gotta love Mathematica sometimes. (Indeed, the function also correctly labels an empty list as strictly decreasing.)

Common Lisp, 43 40 bytes

(defun f(x)`(,(apply'> x),(apply'>= x)))

This takes input as a Lisp list, and returns (T T), (NIL T) and (NIL NIL) to distinguish the 3 categories. Here it is running on the provided test cases:

CL-USER> (mapcar #'f '((7 5 4 3 1)
                       (42 41)
                       (5)
                       (27 19 19 10 3)
                       (6 4 2 2 2)
                       (9 9 9 9)
                       (1 2 3 2)
                       (10 9 8 7 12)
                       (4 6 4 4 2)))
((T T) (T T) (T T) (NIL T) (NIL T) (NIL T) (NIL NIL) (NIL NIL) (NIL NIL))

Jelly, 10 9 5 bytes

-Method found by DrMcMoylex, go give some credit!

;0IṠṀ

TryItOnline! or run all tests

Returns :-1=monotone strictly decreasing; 0=monotone non-increasing; 1=other.

How?

;0IṠṀ - Main link: list
;0    - concatenate with a zero
  I   - incremental differences
   Ṡ  - sign
    Ṁ - maximum

Haskell, 40 bytes

maximum.(zipWith compare<*>tail).(++[0])

Returns

• GT for Monotone strictly decreasing
• EQ for Monotone non-increasing
• LT else

compare compares two numbers and returns GT (EQ, LT) if the first number is greater than (equal to, less than) the second number. zipWith compare<*>tail compares neighbor elements. maximum picks the greatest of the comparison results (where GT > EQ > LT). Before checking a dummy 0 is appended to the input list to make it work for singleton lists, too.

Racket, 44 bytes

(λ(x)(map(λ(p)(apply p`(,@x 0)))`(,>,>=)))

Invoked:

(map (λ(x)(map(λ(p)(apply p`(,@x 0)))`(,>,>=)))
 '((7 5 4 3 1)
   (42 41)
   (5)
   (27 19 19 10 3)
   (6 4 2 2 2)
   (9 9 9 9)
   (1 2 3 2)
   (10 9 8 7 12)
   (4 6 4 4 2)))

Result:

'((#t #t)
 (#t #t)
 (#t #t)
 (#f #t)
 (#f #t)
 (#f #t)
 (#f #f)
 (#f #f)
 (#f #f))

JavaScript (ES6), 51 bytes

a=>a.some((e,i)=>e>a[i-1])+a.some((e,i)=>e>=a[i-1])

Returns 0 for strict decreasing, 1 for non-increasing, 2 otherwise.