How to use "who" vs. "that"

I often get confused when trying to use who vs that.

Some examples that often confuse me:

That

The person that went to the store.

The people that went shopping.

The persons that went shopping.

The group that went shopping.

Who

The person who went to the store.

The people who go shopping.

Please explain when to use either for plural subjects and singular subjects. Animate and inanimate objects as well.

It is usually said that who is used for people (and sometimes animals) while that is used to refer to objects.

In actual usage, though, both who and that can be used to refer to persons, sometimes to animals, and sometimes to entities that consist of people.

• The dog who/that chewed the bone chased the cat.
• The person who/that stole my purse used all my credit cards.
• The group who/that went shopping was mugged.

That, not who, is used to refer to objects.

• The house that Jack built is falling down.

Here's what Oxford Dictionaries Online says:

It is sometimes argued that, in relative clauses, that should be used for non-human references, while who should be used for human references: a house that overlooks the park but the woman who lives next door. In practice, while it is true to say that who is restricted to human references, the function of that is flexible. It has been used for human and non-human references since at least the 11th century, and is invaluable where both a person and a thing is being referred to, as in a person or thing that is believed to bring bad luck.

For more examples of actual usage, here's a link to Merriam Webster's Dictionary of English usage (see pages 895 and 896).

The relative pronoun "that" used in reference to a person or to people has been extant since at the 11th century (I lack proper citation for this date, but there are ample instances throughout 13th and 14th century literature - notably in Chaucer) but is typically only employed when a pronoun is used to combine both a person or people and a thing or things simultaneously.

Merriam-Webster's Collegiate Dictionary, 11th Edition and American Heritage Dictionary both note the interchangeability of "who" and "that" with either being acceptable when referring to a person or to people, however.

(See citations listed with this article.)

According to GrammarBook, there are three basic rules:

1. Who refers to people. That and which refer to groups or things.
2. That introduces essential clauses while which introduces nonessential clauses.
3. If this, that, these, or those has already introduced an essential clause, you may use which to introduce the next clause, whether it is essential or nonessential.

These are the rules as were drilled into my head by both my family and throughout my education. As a matter of personal preference, I would like to add that I find the casual use of that in reference to a person or people to be revealing. To equate someone to a thing is to depersonalize - even when one is referring to one's self.

There isn't any difference for plural and singular subjects; the difference is for people and non-people. Saying who is personifying, saying that or which is objectifying.

Presumably one should always say who when speaking of a human being, or other entity known to be of roughly equivalent or higher intelligence, and that or which when speaking of an inanimate object. Animals and other beings thought to be of below-human cognitive capacity are something of a gray area; I'd say choose your usage according to what sort of tone you want to convey.