There are two reasons your builds might not be starting:
A problem with the trigger set-up
A problem with the webhook
Please note that one git event can only start one build and that the order of your trigger map matters a lot. Learn more about it in Using the Trigger map to trigger builds.
In case that checks out, it’s safe to suspect a problem with the webhook.
Go to your Git repository settings, and look for webhook options.
Make sure you have one registered that starts with hooks.bitrise.io - if it’s there, make sure the scope of the webhook is fitting your use case, meaning you have both pull request and push events selected, if that’s what you want.
If all is well up until this point, you have to look for the responses Bitrise sends to your git provider for these webhooks:
On GitHub and Gitlab, after you’ve clicked Edit, scroll down towards the bottom of the page and select the Responses label at Recent Deliveries. The Body field should contain the reason why your build isn’t triggered. Note that this is a single line of text, not the whole payload.
On Bitbucket, you have to click View Requests on the webhook and Enable history if you haven’t yet, to see this information.