How many verb moods in English? 3,4,5 or more?

Unfortunately, there is no "right information" that can be definitively pointed to. Depending on who you ask, you'll get a different answer.

Some people will list one number, and others will list another. Some people will claim that verbs don't have moods at all.

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It's similar to asking how many verb tenses there are.

From "A Comprehensive Guide to Verb Tenses in English" by Scribendi:

According to some modern linguists, the English language has only two tenses: the past and the present. That's because all of the other tenses are formed by adding auxiliary verbs, such as "be," "do," and "have." In those tenses, the root verb doesn't change; they are not morphological. In other words, because only the past and present tense are, in fact, morphological—"I saw the bear; now, it sees me"—they are the only tenses that count.

Others linguists argue that morphology doesn’t matter. Rather, they stress that the "will" or "shall" one adds to make the future tense—"I will get up in a minute"—has undergone a grammatical shift, rendering it unimportant as a word in its own right when it is used for the purpose of changing tense. In other words, auxiliary verbs are vestigial.

As it so happens, this controversy about the number of verb tenses in English is centuries old. One of the very first grammarians, Ben Jonson, spoke of "three times" in his guide to verb tenses and other grammatical constructions, The English Grammar. According to him, the "times" were the past, present, and future. However, just a century later, another grammarian named Joseph Priestly argued that there were only two tenses and a number of additional "orders."

Among people who are not linguists or grammarians, there is also some legitimate confusion over what constitutes a tense. If you Google "how many verb tenses are there in English," you might get two or three as an answer, but there's an equally good chance you'll get the number 12. That's because people tend to lump tenses together with aspects.

So, are there 2, 3, or 12 verb tenses? Some people say there are 16. And that's not an exhaustive list—I've seen other people give different numbers. It depends on who you ask.

However, the most common answer currently seems to be 12.

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With respect to moods, Bright Hub Education says this in "English Verbs: Tense, Aspect, Mood, and Voice":

Although some grammars identify anywhere between twelve and sixteen English tenses, the nineteen finite, or conjugated, verb forms in English express more than just tense. To be more precise, English has:

• Two tenses: present and past
• Four aspects: simple, progressive, perfect, perfect-progressive
• Three moods: indicative, subjunctive, imperative
• Two voices: active and passive

But if in your world there are more than only two verb tenses, then you might be inclined to reject Bright Hub Education as a source of authoritative information. Also note that if you put all of those things into a chart, you'd end up with 24 entries, excluding the active and passive voice, or 48 entries if you included them.

According to the Wikipedia article, there are at least seven verb moods—with the implication that there are actually more:

In linguistics, grammatical mood (also mode) is a grammatical feature of verbs, used for signaling modality. That is, it is the use of verbal inflections that allow speakers to express their attitude toward what they are saying (for example, a statement of fact, of desire, of command, etc.). The term is also used more broadly to describe the syntactic expression of modality – that is, the use of verb phrases that do not involve inflection of the verb itself.

Mood is distinct from grammatical tense or grammatical aspect, although the same word patterns are used for expressing more than one of these meanings at the same time in many languages, including English and most other modern Indo-European languages. (See tense–aspect–mood for a discussion of this.)

Some examples of moods are indicative, interrogative, imperative, subjunctive, injunctive, optative, and potential. These are all finite forms of the verb. Infinitives, gerunds, and participles, which are non-finite forms of the verb, are not considered to be examples of moods.

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So, how many moods are there? It could be 0, 3, 4, 5, 7, or more. For what it's worth, it seems that the most common answer is 3—as listed at the start of the question. (And, incidentally, as stated by Bright Hub Education—even though they didn't come down on the side of the most common number of verb tenses.)

What is considered "correct" is based on the particular domain vocabulary and mindset of the person answering the question.