- A cheat sheet that shows the most used irregular verbs
- A cheat sheet showing the most used irregular verbs
Both are correct (although neither is a complete sentence, of course; both will become correct full sentences if you put "I would like" at the front of them).
(I have added "the" before "most" to your second version, so that the sentences properly parallel each other. Without the "the", it reads less well.)
The first (...that shows...) has a noun phrase ("a cheat sheet") followed by a subordinate clause (beginning with "that"). The verb of the subordinate clause ("shows") is in the simple present, as you correctly identified.
The second (...showing...) has a noun phrase followed by a participle clause. The verb of the participle clause is "showing". This is not in the present continuous. The present continuous would be "is showing". Here, there is no "is", so it can't be present continuous. This "showing" is a participle but isn't present continuous, nor past continuous, nor future continuous.
A participle clause can be used to postmodify a noun phrase - this is perfectly correct. Examples from Longman's Student Grammar of Spoken and Written English (by D. Biber, S. Conrad, and G. Leech) include:
- families attending the local clinic
- a jeep travelling down Beach Road
- a society consisting of educated people
These can equally well be expressed as follows:
- families who attend the local clinic
- a jeep that is travelling down Beach Road
- a society that consists of educated people
Depending on the context, the actual equivalents might be "...who attended...", "...was travelling..." etc, since the participle clauses work equally well in sentences about the future or past ("I saw a jeep travelling down Beach Road").