Is non-dynamic scalar subquery evaluted for each row?

Is a scalar subquery that’s independent on parent query (does not refer to the parent table) evaluated only once, or for each row, regardless?

SELECT /* ... */
FROM
    t1
WHERE
    parent = (
        -- is this sq evaluated once?
        SELECT
            t2.id
        FROM t2
        WHERE t2.id = 10
    )

or using a CTE

WITH parent_id AS (
    SELECT
        t2.id
    FROM t2
    WHERE t2.id = 10
)
SELECT /* ... */
WHERE
    parent = (
        SELECT p.id FROM parent_id p -- is this sq evaluated once?
    )

If the answer is that the subquery/CTE is evaluated for each row, would joining the CTE on each row improve the performance?

WITH parent_id AS (
    SELECT
        t2.id
    FROM t2
    WHERE t2.id = 10
)
SELECT /* ... */
FROM
    t1, parent_id
WHERE
    parent = parent_id.id

Also, is there a way to figure out from EXPLAIN ANALYZE how many times an expression was evaluated?

Note: the examples are arbitrary and do not reflect real scenario, thus ignore the fact it doesn’t make sense to do a subquery to find out an id which is already known (10).

It is evaluated once.

EXPLAIN ANALYZE will display lines like these:

...
InitPlan 1 (returns $0)
    ->  Index Scan using t2_pkey on t2 (cost=0.29..5.31 rows=1 width=4) (actual time=0.024..0.025 rows=1 loops=1)
          Index Cond: (id = 10)
...

loops=1 tells you the subplan has only been evaluated once. The manual:

In some query plans, it is possible for a subplan node to be executed
more than once. For example, the inner index scan will be executed
once per outer row in the above nested-loop plan. In such cases, the
loops value reports the total number of executions of the node, […]

With a correlated subquery like:

SELECT *, (SELECT id FROM t2 WHERE id = t1.parent_id) AS t2_id
FROM   t1;

(or something more useful) you would see loops=n, where n is the number of rows t1.