She has a perm. (She has a permanent wave. -- Is this the full form?)

She has a polka dot dress.
The girl has pigtails /braids/ plaits. (What is the difference between the three?)
Her older sister is wearing a pink plain dress.
Her brother is wearing a horizontally striped sweater.
She is wearing a vertically orange plain skirt.
The girl is wearing a ribbon on top of her hair.
The girl having a pony tail is wearing a patterned dress.
The wife is wearing a pink polka dot dress.
Her mother has a long wavy black hair.
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Are the expressions all correct? Would you check them?

She has a perm. (She has a permanent wave. -- Is this the full form?) "perm" is the common term; "permanent wave" is the full term but almost no one uses it.

She has a polka dot dress. OK

The girl has pigtails /braids/ plaits. (What is the difference between the three?) These are just different terms for the same thing; "braids" is most commonly used.

Her older sister is wearing a plain pink dress. do you mean "plain" or "plaid"?

Her brother is wearing a horizontally striped sweater. "horizontally" is not usually mentioned

She is wearing a vertically orange plain skirt. delete "vertically" - and do you mean "plain" or "plaid"?

The girl is wearing a ribbon in her hair.

The girl with a pony tail is wearing a patterned dress.

The wife is wearing a pink polka dot dress. OK

Her mother has a<~~delete "a" long wavy black hair. add commas after "long" and "wavy"

Thank you. I mean 'plain' without having colored or patterned shape on the clothing. Is 'plain' commonly used?

Not usually. If there's no plaid or check or pattern or stripe, it's usually not even mentioned.

Braids can be one or more plaited strands of hair and can be wound around the head or worn any way.

Pig tails are usually on small girls and consist of two braids, one hanging down on each side of the head, and they usually have a ribbon tied around the end.

All the expressions you provided are correct. Here's a breakdown and explanation for each of them:

1. She has a perm. - "Perm" is short for "permanent wave," which is a hairstyle that involves chemically or mechanically adding curls or waves to the hair. So, "She has a perm" means that her hair has been treated to have permanent waves.

2. She has a polka dot dress. - This means that the dress she is wearing has a pattern of small, round dots evenly distributed across the fabric.

3. The girl has pigtails / braids / plaits. - Pigtails, braids, and plaits are all hairstyles created by dividing the hair into two sections and tying them separately. The difference lies in the method of tying. Pigtails are simply two ponytails tied at the sides, while braids and plaits involve weaving or intertwining the hair strands. In general, "braids" is often used in American English, while "plaits" is more commonly used in British English.

4. Her older sister is wearing a pink plain dress. - This means that her sister is wearing a dress that is pink in color and does not have any patterns or designs on it. It is a solid, plain color.

5. Her brother is wearing a horizontally striped sweater. - This means that her brother is wearing a sweater with stripes that run horizontally across the fabric, parallel to the ground.

6. She is wearing a vertically orange plain skirt. - This means that she is wearing a skirt that is orange in color and does not have any patterns or designs on it. Additionally, the skirt has vertical lines or texture that run from the waist to the hem.

7. The girl is wearing a ribbon on top of her hair. - This means that the girl has tied a ribbon above or on top of her hair, possibly as an accessory or decoration.

8. The girl having a ponytail is wearing a patterned dress. - This means that the girl, who has her hair tied in a ponytail, is wearing a dress that has a pattern or design on it.

9. The wife is wearing a pink polka dot dress. - Similar to the previous statement, this means that the wife is wearing a dress that is pink in color and has a pattern of small, round dots.

10. Her mother has long wavy black hair. - This means that her mother's hair is long in length, has waves or curls naturally, and is black in color.

All of the expressions you provided are grammatically correct and convey the intended meanings.