My alarm goes off.
Also, you might wanna take a look at this ELU question discussing the why of it.
If you worry that one might perceive a wrong image of "alarm", simply say
My alarm clock went off this morning.
Edit: I find going off for an alarm clock the most viable and common choice. However, the OP seems to be seeking a feasible option among the three words they mention. So, among the three, the correct choice is...drumrolls...
My alarm rings at 7 am.
Proof: I leave it for my first friend to prove it:
The sentence, excluding the adverbial phrase at 7 a.m., returns 15500 results. With the adverbial phrase, it returns 208 results, including some from Google books, which means that the exact sentence is being used by others.
The sentence, excluding the adverbial phrase, returns 1920 results. With the phrase, only one result is found, and it's this very question. Here's why:
1. To set off; initiate. The free dictionary
A lot of what you can make outta the verb trigger goes along the lines of setting something off, activating it. When you want to say a sentence with the context of an alarm waking you up in the morning, this sentence
My alarm triggers at 7 am.
makes no sense. From the grammatical point of view, the sentence is lacking a direct object. "Your alarm triggered what, exactly?"
From the "meaning" view, trigger means to activate. It's not you who activates the alarm, because you're sleep. However, the meaning implies that this sentence is sensible:
My mobile device has the functionality of triggering an alarm to wake me up every morning.
Excluding the adverbial phrase, there are some 26k results in Google. So, the phrase my alarm bells is very common. However, with the adverbial phrase, there are only 2 results of Google: One directing to this question and the other one to "ELL StackExchange - Newest questions". Here's why:
1. To put a bell on.
2. To cause to flare like a bell.
To assume the form of a bell; flare. Free Dictionary
The verb form of bell is used for a very different purpose than what you had in mind. Bells (i.e. the noun form) are instruments that are (or rather were) used to alarm people. They look something like this:
This is a musical instrument.