Pilot studies are conducted for various purposes. Firstly, it could be a mini-scale feasibility study to explore the bottlenecks, test research protocol and observe any tentative problem that may arise during the main study. This mini-scale study may also identify the sample variance (to estimate the population variance) and aid in sample size determination for the main study. It may also be aimed at validating measurement instruments (e.g., questionnaire) and helps in facilitating the main study by looking at whether the questions are yielding responses in a consistent way.

The literature supports various figures of sample sizes depending on the purpose of the main study. For pilot studies, there is a ubiquitous suggestion to use a sample size which is 10% of the sample size of the proposed main study (reference 1). So if your main study plans to sample 500 respondents, you may choose 50 for your pilot study. (A general caution is that you may not include the pilot study responses in analyzing the responses of the main study, unless there is extreme or extra-ordinary circumstances where the sampling subjects are extremely scarce or rare. Interestingly, there are also some studies that recommend to include pilot study samples in the main collection of samples. This is altogether a different discussion).

Other than the ubiquitous 10% rule of thumb, literature also suggest a variety of pilot study sample sizes depending on its objective. According to reference 2, for a feasibility study, samples as small as 10-15 per group may be sufficient. For instrument development and testing, the article recommended pilot testing with 25 to 40 respondents per group. A sample size of 20 to 25 is mentioned for pilot testing for intervention efficacy, provided reasonable effect sizes are achieved. A sample size of 30 to 40 per group were recommended for pilot studies comparing groups. For social science research, the reference 3 found that a representative minimum sample size of 30 could be sufficient for a pilot study, irrespective of the sample size of the main study. The article also mentioned about minimum pilot sample size of 12 per group, in studies where two or three groups might be expected.

Hope this answer helps you.

References:

1. Lackey, N.R., & Wingate, A.L. (1998). The pilot study: One key to research success. In P.J. Brink & M.J. Wood (Eds.), Advanced design in nursing research (2nd ed.). Thousand Oaks, CA: Sage.

2. Hertzog, M. A. (2008). Considerations in determining sample size for pilot studies. Research in nursing & health, 31(2), 180-191.

3. Johanson, G. A., & Brooks, G. P. (2010). Initial scale development: sample size for pilot studies. Educational and Psychological Measurement, 70(3), 394-400.

It depends on your purpose but I usually use %10 of sample size or at least 30. Also if you'll use special statistical analyses, the pilot should let you at least validate the assumptions for that analyses.

A lot depends on your goal. If you are testing a new measure, such as a scale, then you will need enough observations to get reasonable estimates of the correlations involved. This would suggest an N = 50 or so.

Alternatively, if you just want to assess the effectiveness of your question wording, then a set of 10 or so cognitive interviews would be sufficient.

It depends on the purpose of the pilot study. however, it should not be less than 40.