Psalm 97 is in the midst of the Enthronement Psalms (93-99). At the beginning (93:1) middle (96:10; 97:1) and end (99:1) of the unit is the declaration, “The Lord reigns.” His throne is an eternal throne (93:2). He is the judge of the earth (94:2). He is the great King above all gods (95:3; 97:9) who is greatly to be praised (96:4). His sanctuary is one of splendor and majesty (96:6) where he is worshipped as King with trumpets and horns (98:6).
The simplest way to divide Psalm 97 is right in half. Artur Weiser sees verses 1-6 describing the theophany of the Lord while verses 7-12 describe the results of that theophany. Such a division is true to the text and I would only make the very modest revision of identifying verses 1 and 12 as prescript and postscript enjoining praise to the appearing Lord.
More commentators see three divisions in the text. W. Stewart McCollough and James L. Mays identify 1-5; 6-9; 10-12 as sections while Mitchell Dahood follows the same but includes verse 6 with the first instead of second section.
Willem VanGemeren goes a step further and sees a four-part chiastic structure of:
A: The revelation of YHWH’s glory (1-6)
B: Exhortation to worship (7)
B’: Zion’s Worship (8-9)
A’: The effects of YHWH’s glorious rule (10-12)
As far as the over-all structure I think Weiser’s fits best with what is going on the text. Yet seeing the evident symmetry of verses 1 and 12 I pursued the possibility of a more developed chiastic structure and arrived at the following:
1 The LORD reigns, let the earth rejoice; let the many coastlands be glad!
2 Clouds and thick darkness are all around him;
righteousness and justice are the foundation of his throne.
3 Fire goes before him and burns up his adversaries all around.
4 His lightnings light up the world; the earth sees and trembles.
5 The mountains melt like wax before the LORD,
before the Lord of all the earth.
6 The heavens proclaim his righteousness,
and all the peoples see his glory.
7 All worshipers of images are put to shame,
who make their boast in worthless idols;
worship him, all you gods!
8 Zion hears and is glad, and the daughters of Judah rejoice, because of your judgments, O LORD.
9 For you, O LORD, are most high over all the earth;
you are exalted far above all gods.
10 O you who love the LORD, hate evil! He preserves the lives of his saints;
he delivers them from the hand of the wicked.
11 Light is sown for the righteous, and joy for the upright in heart.
12 Rejoice in the LORD, O you righteous, and give thanks to his holy name!
Some brief notes concerning the pairings:
Verses 1+12 A call to rejoice in the Lord
Verses 2+11 Contrast of light and darkness; righteousness is foundation of Lord’s throne and light and joy are given to the righteous
Verses 3+10 The Lord destroys his adversaries and protects his saints
Verses 4,5+9 All the earth trembles before the Lord who is exalted over all the earth
Verses 6+8 Heaven proclaims and the righteous hear
Verse 7 Worship the true Lord
Some issues with this structure:
The pairing of verses 4 and 5. I kept these two verses together because of the shared themes of “the world,” “the earth,” “all the earth;” and the similar ideas of “trembling” and “melt like wax.”
It is very tempting to make verses 7 and 8 as the joint-center since they contrast the response of the wicked and the righteous to the appearing of the Lord.
 Marvin Tate also sees two divisions, but identifies them as 1-9 and 10-12.