Neither is used for two things.

• Neither of my parents is retired.

None is used for three or more things.

• None of the kids in the class failed the test.

About singular or plural, this depends.

Technically, after neither, we should always use the singular, as in the example above (Neither of my parents is retired / Neither my mom nor my dad is retired.)

However, some people do use the plural: Neither of my parents are retired; Neither of the athletes were injured.

After none, use the plural if the object is countable:

• None of the facts are correct.

Use the singular if the object is uncountable:

• None of the information is correct.