1. He went out for a run.

2. He went out for a ride.
(for riding a horse or riding a bike?)
3. He went out for a skate/a ski.
4. He went out for a smoke.
5. He went out for a drink.
6. He went out for a tour.
7. He went out for _________.
8. She went out for a shop.
9. She went out for a dance.
10. She went out for a meeting.
11. He went out for a shopping.
12. He went out for a buying.
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Are they all grammatical? Can we use almost all the expressions in the blank in #7?

1. He went for a run.

2. He went for a ride.
(for riding a horse or riding a bike?)
3. He went for a skate/a ski.
4. He went for a smoke.
5. He went for a drink.
6. He went for a tour.
7. He went for _________.
8. She went for a shop.
9. She went for a dance.
10. She went for a meeting.
11. He went for a shopping.
12. He went for a buying.
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Are they all grammatical? Can we use almost all the expressions in the blank in #7?

In both lists ...

1 and 2 are fine.
3 is not correct.
(He went skating or He went skiing.)
4, 5, 6, and 10 are fine.
7 will sometimes work, but sometimes it won't.

She/He went shopping.
She/He went out shopping.
She/He went out dancing.

Be careful of 7 -- there are many variations, as you can see.

1. He went out for a run. (Correct)

2. He went out for a ride. (Can be used for both riding a horse or riding a bike)
3. He went out for a skate/a ski. (Correct)
4. He went out for a smoke. (Correct)
5. He went out for a drink. (Correct)
6. He went out for a tour. (Correct)
7. He went out for _________. (Yes, you can use various activities in the blank depending on the context. For example, you could use "He went out for a hike," "He went out for a swim," "He went out for a movie," etc. The choice of activity will depend on what the person is doing or what they plan to do outside.)
8. She went out for a shop. (It's more common to say "She went out to shop" or "She went out shopping.")
9. She went out for a dance. (Correct)
10. She went out for a meeting. (Correct)
11. He went out for a shopping. (It would be more appropriate to say "He went out shopping.")
12. He went out for a buying. (It would be more appropriate to say "He went out to buy something.")

So, not all the expressions are grammatically correct, but they can be easily corrected by using the appropriate form.