Does secular Buddhism have a set of texts that are particularly relevant to it. I can imagine that there are texts that won't have any relevance to secular Buddhists such as the twin miracles. But are texts that have particular resonance even insofar as developing in a set of texts that could almost be described as a canon (in the making maybe?)
Historical Buddhist scholars trying to figure out who the historical Buddha was prefer using the Pali texts, not so much because they are canon but because they pass the various tests historians use to decide how old a text is and appear to be the oldest and closest to the original man. This isn't though a secular project, it appears to be a project that secular Buddhists seem to like. As far as we can tell, the historical Buddha was more secular than your average guy and he was a historical person. Personally, I'm not to excited about this project because immediately after the historical Buddha put forth his project, he or people around him mixed in supernatural elements.
Stephen Bachelor seems to have a preference for Pali texts, since the tend to be less devotional and more straight forward. So his project is one of taking the Pali texts and kicking out the supernatural.
David Chapman is a blogger an Meaningness and uses the tantric tradition as his starting point. Despite tantra sometimes being called Indian witchcraft, if you kick out the supernatural you have the core for a secular religion. This project would be drawing on the Tantric texts of ~800AD.
Chögyam Trungpa started with Vajrayana and wrote his own terma. Canon is a matter of institutions and supernatural beliefs about old texts. So why not write a modern text?
D. T. Suzuki took Zen and ignored all the supernatural elements to create DT Suzuki style Zen. I guess the material the DT Suzuki would be canon.
So I think this illustrates that secular Buddhism is sort of a methodology where you pick a flavor of Buddhism and kick out the supernatural, add some modern thinking from science and elsewhere. Secular Buddhists have been picking a variety of "starting points". I chose Chinese Buddhism-- the big themes in Huayen & Chan speak to me and are still there when I remove the devotionalism and supernatural.
Some of these secular Buddhists at the moment are just successful authors and teachers, Chögyam Trungpa has an institution that outlived him, although the word is Shambhala is less secular that it used to be.
I think it varies tremendously based on what tradition they are adapting their teachings from. People following a secularized vipassana tradition probably uphold the Satipatthana sutta, and secular Zen practitioners probably take interest in the Heart Sutra, the Diamond Sutra, and the Platform Sutra.
From my interactions with secular Buddhists I get the impression that when they do study canonical texts they don't have a very wide selection, just a few that they consider important.