CREATE TABLE jsonthings(d JSONB NOT NULL);
INSERT INTO jsonthings VALUES
    ('{"name":"First","tags":["foo"]}')
  , ('{"name":"Second","tags":["foo","bar"]}')
  , ('{"name":"Third","tags":["bar","baz"]}')
  , ('{"name":"Fourth","tags":["baz"]}');
CREATE INDEX idx_jsonthings_name ON jsonthings USING GIN ((d->'name'));

EXPLAIN ANALYZE SELECT d FROM jsonthings WHERE d->'name' ? 'First';

                                                            QUERY PLAN
----------------------------------------------------------------------------------------------------------------------------------
 Bitmap Heap Scan on jsonthings  (cost=113.50..30236.13 rows=10000 width=61) (actual time=0.024..0.025 rows=1 loops=1)
   Recheck Cond: ((d -> 'name'::text) ? 'First'::text)
   Heap Blocks: exact=1
   ->  Bitmap Index Scan on idx_jsonthings_name  (cost=0.00..111.00 rows=10000 width=0) (actual time=0.015..0.015 rows=1 loops=1)
         Index Cond: ((d -> 'name'::text) ? 'First'::text)
 Planning time: 0.073 ms
 Execution time: 0.047 ms
(7 rows)

EXPLAIN ANALYZE SELECT d FROM jsonthings WHERE jsonb_exists(d->'name','First');
                                                            QUERY PLAN
----------------------------------------------------------------------------------------------------------------------------------
 Seq Scan on jsonthings  (cost=10000000000.00..10000263637.06 rows=3333334 width=61) (actual time=0.016..3135.119 rows=1 loops=1)
   Filter: jsonb_exists((d -> 'name'::text), 'First'::text)
   Rows Removed by Filter: 10000003
 Planning time: 0.051 ms
 Execution time: 3135.138 ms
(5 rows)

CREATE OPERATOR ### (
  PROCEDURE = jsonb_exists,
  LEFTARG = jsonb,
  RIGHTARG = text,
  RESTRICT = contsel,
  JOIN = contjoinsel);

EXPLAIN ANALYZE SELECT d FROM jsonthings WHERE d->'name' ### 'First';
                                                           QUERY PLAN
--------------------------------------------------------------------------------------------------------------------------------
 Seq Scan on jsonthings  (cost=10000000000.00..10000263637.06 rows=10000 width=61) (actual time=0.012..3381.608 rows=1 loops=1)
   Filter: ((d -> 'name'::text) ### 'First'::text)
   Rows Removed by Filter: 10000003
 Planning time: 0.046 ms
 Execution time: 3381.623 ms
(5 rows)

Name        Indexed Data Type   Indexable Operators
...
jsonb_ops   jsonb               ? ?& ?| @>

CREATE OR REPLACE FUNCTION jb_contains(jsonb, text)
  RETURNS bool AS
'SELECT $1 ? $2' LANGUAGE sql IMMUTABLE;

CREATE INDEX idx_jsonthings_name ON jsonthings USING GIN ((d->'name'));

CREATE INDEX idx_jsonthings_d_gin ON jsonthings USING GIN (d);

SELECT d FROM jsonthings WHERE d @> '{"name":"First"}'

CREATE INDEX idx_jsonthings_d_email ON jsonthings ((d->>'email'));

SELECT d FROM jsonthings WHERE d->>'email' = 'First';