With examples from Psych.
Simple answer: You never use "alright," because it isn't accepted as a real word. Despite the growing usage of the spelling "alright" and the fact that it's been showing up in written records for over a century, "alright" is considered a misspelling of "all right."
So that's easy enough. Whenever you want to write "alright," just write "all right" and you can never go wrong. This is true for any dialect of English.
Of course, in the real world, it's a little more complicated. The issue of "all right" vs. "alright" and whether the latter should be accepted as its own word has been the subject of fierce debate among grammar experts for as long as the word has been around. So let's take a closer look at the term.
The phrase "all right" can be used as an adjective or an adverb, and has two major definitions.
1. Acceptable, agreeable, good, okay.
"I have an idea, Spencer," said Lassiter. "You should let the actual police handle this. That all right with you?"
2. All correct.
Shawn smirked, certain that the numbers he'd seen in his "vision" were all right.
The spelling of "alright" most likely comes from conflating "all right" with phrases of similar construction; namely, "all together" and "all ready." Both of these phrases are similar to words ("altogether" and "already") that are distinct and not interchangeable with the two-word phrase. It's not hard to see why people would think "all right" fits into that group and "alright" must therefore be a word.
This is probably a case of language evolving. Because of the conflation with similar words, "alright" is usually thought to have a different definition from "all right," just like "altogether" and "all together" have separate meanings. Many writers typically mean to give "alright" the "acceptable, agreeable, good" definition above, and reserve "all right" for the "all correct" definition.
"Buddy, you're just going to have to trust me, alright?" said Shawn.
It seems that the spelling "alright" is becoming more accepted (especially, according to one of my sources, in British English), and if you use it in writing, chances are most people won't look at you askance. But it is technically still considered a misspelling of "all right," so it would probably be easiest to just use that spelling every time. Certainly, use "all right" for all formal or academic writing.