שמם

Abarim Publications' online Biblical Hebrew Dictionary

שמם ישם שמים

The two roots שמם (shamem) and ישם (yasham) are obviously related in form and meaning. More curious is the root שמה (smh):

The verb שמם (shamem) means to be desolate (of roads or regions) or to be appalled. Although BDB Theological Dictionary declares the connection between these two different meanings "not clear," HAW Theological Wordbook of the Old Testament reports a basic meaning of this root of "desolation caused by some great disaster, usually as result of divine judgment" (2 Samuel 13:20, Leviticus 26:22, Daniel 11:31, Ecclesiastes 7:16).

This root's derivations are:

The adjective שמם (shamem), meaning devastated or deserted (Daniel 9:17 and Jeremiah 12:11 only).
The feminine noun שממה (shemama), meaning devastation or waste (Exodus 23:29, Leviticus 26:33). The Masoretes interpreted the pronunciation of this word in Ezekiel 35:7 slightly different as shimema. HAW treats it as separate word but BDB doesn't.
The feminine noun שמה (shamma), meaning waste (Hosea 5:9, Isaiah 5:9), or horror or appalment (Deuteronomy 28:37, Micah 6:16).
The masculine noun שממון (shimmamon), meaning horror (Ezekiel 4:16 and 12:19 only).
The feminine noun משמה (meshamma), meaning devastation (Ezekiel 6:14) or horror (Ezekiel 5:15).

ישם

The verb ישם (yasham) means to be desolate, predominantly of dry, arid country (Genesis 47:19, Ezekiel 12:19). Its derivatives are:

The feminine noun ישימה (yeshima), which in theory should mean desolation, but it's not wholly clear whether this word actually exists. It may occur in Psalm 55:15 only, in the plural form ישמות (yeshimot; Desolation upon them!) but ancient Jewish commentators thought it should probably read ישיא מות (yesha' mot), meaning: Let death be. Most modern translations go with the corrected version.
The masculine noun ישימון (yeshimon, or variants), meaning wilderness or waste land (Numbers 21:20, Deuteronomy 32:10, Psalm 68:7), which occurs frequently juxtaposed by the word מדבר (midbar), also meaning desert. Our noun often occurs preceded by the definite article ה (he), and some interpreters take it to be a proper name, (the) Jeshimon, comparable to (the) Negev and (the) Arabah.

שמה

The assumed root שמה (smh) isn't used as verb in the Bible, and its sole derivative is the important masculine noun שמי (shamay). It occurs only in plural: שמים (shamayim) and means heaven(s). It's not clear where this word comes from, and thus what exactly the concept of heaven meant to the Hebrews, but it bears a striking resemblance to the above verbs. It goes against common intuition to interpret the heavens as place of destruction, but a similar association exists in the divine names Shaddai (may mean My Destroyer) and YHWH (may mean He Will Cause To Fall).

Our word שמים (shamayim) also bears a striking resemblance to the noun שם (shem), which means name or renown and which is also a word of mysterious origin. But note the obvious connection between going to heaven and making oneself a name, reflected in the story of the tower of Babel (Genesis 11:4).

Our word שמים (shamayim) appears to describe a concept that contrasts ארץ ('eres), which is usually translated as earth or land, but which in fact conveys one's present position, the present state of affairs and present state of knowledge. Land was considered to come from water (מים, mayim, which makes water representative of earth's unapplied, raw material), whereas the heavens represented earth's potential; the ultimate limit to which it could grow. Only when earth would have grown to its fullest potential, earth and heaven could coexist (Revelation 19:7), but until then one of the functions of the heavens was to release water on the earth so that the present situation could either erode and fade away, and make room for a new situation, or else so that the present situation could continue to grow (and note that the name Torah is closely related to a word meaning rain).

In the Bible, שמים (shamayim) is a place or situation that people are designed to properly work towards and intuitively desire to go (Genesis 3:16, Haggai 2:7, Philippians 3:20). It's where God lives (Deuteronomy 26:15, Isaiah 66:1), where his angels and other agents of guidance have their headquarter (Genesis 1:14-18, Matthew 22:30), and where rain and dew come from (Genesis 27:28, Judges 5:4), as well as destructive fire (2 Kings 1:10).