שער

Abarim Publications' online Biblical Hebrew Dictionary

The Hebrew of the Bible contains an enormous array of words of or related to the form שער (s'r). And since the distinction between שׂער (s'r; dot to the left) and שׁער (sh'r; dot to the right) was made more than a thousand years after the Bible was written, there was no way of telling which of the roots was expressed in the text.

Linguists identify four separate roots שׂער (s'r) in the Bible, but closer scrutiny reveals that all these four roots can be easily united in a single basic meaning. Then there is one root, which is spelled סער (s'r) and which is an alternate form of one of the roots שׂער (s'r). And then there are three more roots שׁער (s'r), which can also be shown to be of the same or similar intent.

The most basic idea captured by all these roots (or, dare we say, there is only one root with a huge amount of derivations and applications), is an intense emotional experience. For an exhaustive investigation of these seven roots, read our article on hair in the Bible.

שׂער I

The root שער (s'r I) isn't used in the Bible and because it isn't used we also don't know what it may have meant. But in cognate languages it exists with the meaning of being hairy. And there are quite a few derivatives of this root extant from which to attempt to distil a common charge:

The masculine noun שער (se'ar) means hair in general (Genesis 25:25, Judges 16:22)
The feminine noun שערה (sa'ara) denotes a single hair (1 Samuel 14:45).
The curious denominative verb שער (sa'ar) means to be very afraid. What this verb has to do with the previous nouns is not immediately obvious, but HAW Theological Wordbook of the Old Testament proposes that it might literally mean to bristle with terror. But that's far from all. In the Bible there seems to be a certain symbolic meaning to hair, something that is completely lost in modern languages and translations (for more details, read our extensive study on hair in the Bible, mentioned above).
The masculine noun שער (sa'r) derived from the verb, meaning horror (Job 18:20 only).
The adjective שעיר (sa'ir) means hairy (Genesis 27:11 and 27:23 only).
The masculine noun שעיר (sa'ir) denotes a he-goat (a bristly guy; Leviticus 4:24). Older translations of the Bible (and even the venerable BDB Theological Dictionary) have chosen to translate this noun with demon or satyr in four occasions (Leviticus 17:7, 2 Chronicles 11:15, Isaiah 13:21 and 34:14). Later interpreters feel "no need to go to this bizarre translation" (in the words of HAW Theological Wordbook of the Old Testament).
The feminine noun שעירה (sa'ira) means she-goat (Leviticus 5:6).
The feminine noun שערה (se'ora) means barley, the bearded grain (Exodus 9:31, Isaiah 28:25).

שׂער II

The verb שער (sa'ar II) means to sweep or whirl away, usually in relation to a storm wind (Psalm 58:9, Job 27:21 - and hence the curious image of our own culture of "(buck-)riders on the storm" ). This verb's derivatives are:

The masculine noun שער (sa'ar), meaning storm (Isaiah 28:2 only).
The feminine noun שערה (se'ara), also meaning storm (Nahum 1:3 and Job 9:17 only).

שׂער III

The verb שער (s'r III) probably means to be acquainted with, or shudder in the face of something (as proposed by the renowned linguist Gesenius). It's used only once, in Deuteronomy 32:17, " . . . to gods whom they have not known . . . " (New American Standard).

שׂער IV

The assumed root שער (s'r IV) has as sole derivative the masculine plural noun שעירים (se'irim), meaning raindrops. This noun is used only once, in Deuteronomy 32:2.

סער

Root שׂער (s'r) is really nothing but an alternate spelling of the verb סער (sa'ar), which also means to storm or rage. It's not clear if this verb is denominative, but either its derivatives or parent nouns are:

The masculine noun סער (sa'ar), meaning storm or tempest (Psalm 55:8, Amos 1:14).
The feminine noun סערה (seara), also meaning storm or tempest (2 Kings 2:1, Zechariah 9:14).

שׁער I

The assumed root שער (sh'r I) occurs in cognate languages with the rather violent meaning of to break off or through, to tear in two, split, divide or tear down. In the Hebrew Bible two derivations remain:

The masculine noun שער (sha'ar), meaning gate; "the means of controlled access to a walled city" in the words of HAW Theological Wordbook of the Old Testament. This noun occurs all over the Old Testament (Genesis 22:17, Exodus 32:26, Isaiah 14:31).
The masculine noun שער (sho'er), meaning gatekeeper or porter (2 Samuel 4:6, 2 Chronicles 31:14).

שׁער II

The verb שער (sha'ar II) means to calculate or reckon (Proverbs 23:7 only). Its sole derivative is the masculine noun שער (sha'ar), meaning measure (Genesis 26:12 only).

שׁער III

The assumed root שער (sh'r III) yields the following derivatives:

The adjective שער (sho'ar), meaning horrid or disgusting. It occurs only once, in Jeremiah 29:17.
The feminine noun שערורה (sha'arura), meaning horror or horrible thing (Jeremiah 5:30 and 23:14 only).
The feminine noun שערוריה (sha'aruriya) or שעררית (sha'arurit), also meaning horrible thing (Hosea 6:10 and Jeremiah 18:13 only).