How many basic, deadly, primary, or great fears are there? Is public speaking one of them?

Well, that depends on which wise man you ask.

In their book Repacking Your Bags: Lighten Your Load for the Good Life, Richard Leider and David Shapiro (3rd edition, 2012) list just Two Deadly Fears on page 125:

1. Fear of Not Having Enough
2. Fear of Not Being Enough

An article in the Spring 2009 issue of the Buddhist magazine Tricycle by Ezra Badya described and discussed The three things we fear most, which were:

1. Losing safety
2. Aloneness and disconnection
3. Unworthiness

In the earlier 2nd edition of their book Repacking Your Bags: Lighten Your Load for the Rest of Your Life, Richard Leider and David Shapiro had listed Four Deadly Fears on page 33:

1. Having lived a meaningless life
2. Being alone 
3. Being lost
4. Dying

The third paragraph in the introduction of Jamal Yogis’s 2013  book The Fear Project says:

“The Buddhist Avatamsaka Sutra emphasizes that to become a bodhisattva, an enlightened being full of compassion, one must transcend the five primary fears: loss of livelihood, bad reputation, death, negative rebirths, and stage fright. The Buddha apparently wouldn’t have been surprised that public speaking still often tops fear polls.”

In his 1937 book Think and Grow Rich Napoleon Hill (the emperor of self-help) said that the Six Basic Fears are:

1. Poverty
2. Criticism
3. Ill health
4. Loss of love of someone
5. Old age
6. Death

The Buddhist list of five primary or great fears is the only one that explicitly includes stage fright (sometimes translated instead as fear of speaking before a public assembly). On other lists like Ezra Badya’s it is implicit, like losing safety and unworthiness. 

The image is a recolored version of a poster from 1900 for The Evil Eye.