I agree with the suggestion user 3169 gave: One time out of ten it is a commonly used expression (see definition 14 in LDOCE definition of time)
Your other suggestion to phrase it sounds good:
It happens once every ten times.
It is used for something that happens repeatedly, sometimes used for regular events(see def. 2).
There is evidence of usage (which I found in a Google books search).
I found some examples of usage for:
once in 10 times
but these were mostly for probability, and you want to express frequency or a proportion.
Your suggestion in comments:
10% of the time
or, slightly rephrased
in 10% of the cases/attempts
also sound good.
**Side note: the use of present simple indicates a repeated problem. If you use past simple in combination with on(c)e out of 10 to say: It happened once, out of 10 times or It happened one time out of 10 this would indicate that you have tried 10 times and encountered this problem only once.On the other hand once every n times indicates that the problem happened repeatedly, regardless of the verb tense you use.