kapisse (kappi) = kapi sisse 'into the cabinet'
kapis = kapi sees 'in the cabinet'
kapist = kapi seest 'out of the cabinet'
lauale = laua peale 'onto the table'
laual = laua peal 'onto the table'
laualt = laua pealt 'onto the table'
A native speaker, Myst, points out that the 'long illative' kapisse has fallen out of use in the modern spoken language, in favour of the 'short illative' kappi. That gives me another idea for a future post when I look at cases, the 'short' v. the 'long' illative. Thanks Myst!
Note: The observent student will notice that the case endings -sse, -s, -st, -le, -l, -lt have jumped from the noun to the postposition. For example, peale, peal, and pealt simply mean 'onto the head', 'on the head', and 'off the head' respectively. The Estonian word for head is pea. It is related to the Finnish and Võro words for head pää as in the Southern Estonian town of Otepää (Otõmpää in Võro) which means 'Head of the Bear'.
A native speaker, Kata, adds the following: 'There certainly is a historic connection with the noun 'pea' and the postpositions you named. Today the postposition have only one meaning. For example 'onto the head' would be 'pea peale': Ta elu pöörati pea peale. [Her/His life turned onto the head (lit. Her/His life turned upside-down.]
And one more historical note. I am currently reading a Estonian translation of a novel, published in 1937. "Ta vaatas tema peale" [S/he looked him in the face (lit. onto the head)] is a common way of expression. The case has shifted within the 70 years and now contemporary Estonians would say: "Ta vaatas teda." [S/he looked at her/him]' Thanks for the input Kata!
Here are some examples of the Estonian locative cases and postpositions in action:
Karu tuleb koopa seest [= koopast] välja 'The bear comes out of the cave'.
Ma mõtlen sõbra peale [= sõbrale] 'I am thinking of my friend'.
Räägi kõik südame pealt [= südamelt] ära! 'Tell everything that is on your heart [Get everything off your chest!]'.
Vean kihla kümne dollari peale [= kümnele dollarile] 'I will bet you ten dollars'.
Must valge peal [= valgel] 'Black on white'.
All note the use of käsi 'hand' as a postposition:
Kelle käes [= Kellel] on minu ajaleht? 'Who has my newspaper? [In whose hand is my newspaper?]'
Ma küsin venna käest [= vennalt] 'I will ask (of) my brother'.
Source: Estonian Textbook by Juhan Tuldava, 1994, pg. 100.