These acrostics can be quite useful to us. For example, the psalms numbered 9 and 10 in the Hebrew Bible were originally one psalm. We know this because the verses form an alphabetic acrostic. The two parts appear together in the Septuagint (LXX), the Greek translation of the Hebrew Bible that was done in Alexandria in the second century BCE. They appear separately, though, in the official Masoretic Hebrew text, and that is why we have two numbering systems for the Psalms.
Psalm 110 (111) and Psalm 111 (112) are both alphabetic acrostics having very short lines. This makes their versification certain, and allows us to study Hebrew poetic meter. Similarly, Psalm 118 (119) is a unique alphabetic acrostic in which each section (there are eight lines per section) begins with a letter of the alphabet. The Hebrew alphabet has twenty-two letters. and 8 x 22 = 176, which is the total number of lines in this psalm, which clarifies the versification and gives us an even larger sample for the study of Hebrew metrics.
Finally, in Psalm 144 (145) the verse corresponding to the letter "Nun" is missing entirely from the official Masoretic text. This has given rise to all sorts of conspiracy theories concerning the missing verse. Fortunately for us, the missing verse is still found in the Greek of the LXX, in the Syriac of the Peshitta (ca. 500 CE), and in both of Jerome's Latin translations of the Psalms, the first of which is based on the Greek of the LXX, and the second of which is based on a Hebrew text of his day (ca. 400 CE), which was evidently in better condition than the official Masoretic text. It's not difficult to turn the Greek or the Syriac or the Latin back into Hebrew, and when you do, it does, indeed, begin with "Nun." As to the conspiracy theories, they are quite unnecessary. The missing verse translates to: "The Lord is faithful in all his words, and holy in all his works."
What about non-alphabetic acrostics? Well, I haven't yet found any in the Psalms. I would think the most likely text to spell out would be the Shema. It has twenty-five letters, so the psalm would have to have that number of verses. Happy hunting.
