prop.test or chi squared test on count data with 3 groups? I have some count data pertaining to number of events observed among independent trials in 3 groups of an experiment: 
count   A    B    C
0       5    0    5
1       1   25    9
2       9   30   10
3       3   15    8
4       4   13    9
5       4   13    5
6       1    6    8
7       2    4    6
8       0   10    1
9       4    5    2
10      0    5    3

A Kruskal-Wallis test leads me to conclude that the distribution of numbers of events observed does not differ significantly between groups. However, I'd also like to know whether getting a specific event count is statistically more likely in the different groups. In particular I'm interested in event counts of zero or 1. For example, an event count of 1 is proportionately much more likely in group B compared with groups A or C. I'd like to test for the significance of this.
I have looked at using a chi-squared test to compare proportions between groups for a specific outcome but believe this may be invalid due to low frequencies of observation (<5) for some numbers of events. In this case, would R's prop.test be more suitable?
EDIT: post updated to make data easy to copy-and-paste
 A: You are mainly interested in knowing how to interpret the test results when there are more than two categories. Particularly, as you stated: "I would like to know whether getting a specific event count is statistically more likely in the different groups"
Therefore, I am sharing a brief script that I use in such situations:
# Load the required packages:
library(MASS) # for chisq
library(descr) # for crosstable

CrossTable(a$exam_result, a$ethnicity
       fisher = T, chisq = T, expected = T,
       prop.c = F, prop.t = F, prop.chisq = F, 
       sresid = T, format = 'SPSS')

This code will generate both Pearson's Chi-square and Fisher's Chi square. It produces counts as well as proportions of each of the table entries.
Based on the standardised residuals or z-values scores i.e., 
sresid

For each table entry, look at its sresid, whenever it is outside the range |1.96| i.e., less than -1.96 or greater than 1.96, then it is significant p < 0.05. 
And its sign would then indicate whether it is positively related or negatively.
