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Using a Fligner test to infer about the respect of the assumption of homoscedasticity is not very smart given that the Fligner test tests to the null that there is no difference of variance between the groups. This will wrongly favors small sample size. As it has been said by @Michael Mayer here.

How can we further investigate if the assumption of homoscedasticity is respected?

Is it worth plotting the model's residuals versus the fitted values? Below lines are R coded:

m = aov(myFormula, myData)
plot(residuals(m), m$fit)

I don't have much experience in statistics and it seems rather hard for me to decide from this plot whether the assumption of homoscedasticity is respected. What else can I do?

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Remi.b
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    If you `plot(m)` you should get 4 plots, including a scale-location plot, which should be easier to interpret - if the mean of the scale location plot clearly changes, you don't have homoskedasticity. – Glen_b Nov 20 '13 at 01:37

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You might consider Levene's test, and this discussion of how to use it in R. It may be less sensitive to non-normal distributions than Bartlett's test, according to the following reference. There's also the Brown-Forsythe test for ANOVAs on transformed response variables, though this may be another version of Levene's test, judging from this question.

It sounds like your independent variable (IV) has discrete categories, but if this isn't the case, this answer provides loads of good recommendations for testing homoscedasticity across a continuous dimensional IV.

Reference

Bartlett's test. NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.7. Available online, URL: http://www.itl.nist.gov/div898/handbook/eda/section3/eda357.htm. Retrieved December 31, 2013.

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Nick Stauner
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    You are correct about the relationship between Levene's and Brown-Forsythe--they both use the exact same formula, but the latter uses the median, and the former uses the mean. https://en.wikipedia.org/wiki/Brown%E2%80%93Forsythe_test – jeramy townsley Jan 27 '16 at 05:08