Is there any way to generate a report that lists a count of the labels applied to the cards in a given list? So in other words, having a report showing lists as rows, labels as columns, and a count value in the fields.
Unfortunately, no such report exists from Trello! One option is the JSON export of a board, but that data can certainly be a little difficult to parse, though the information is in there.
Alternatively, if you have Trello Business Class, you also have the option of a CSV export, which is organized a little differently than you described—there, each card is its own row, and if there's a label on the card, that will be listed in a column.
More information on these options is available here: http://help.trello.com/article/747-exporting-data-from-trello-1
You may also want to look into the Trello API, which will let you poll for whatever data you need! https://developers.trello.com/
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