Spirituality
Spirituality is defined here as the personal pursuit of one's core beliefs and connections to the universe, and attempts to become more mentally attuned to those beliefs and connections. It is separated fully from belief in the supernatural in order to create a "secular spirituality" that is compatible with science.
It's All In Your Brain
Since the source of all morality, and indeed consciousness itself, is the brain, most "spiritual" pursuits involve attempts to become more in touch with the internal reward systems of our brains.
Buddhism is a spiritual tradition that is almost completely psychological in terms of its practices.
Spiritual tradition is ritualized mental health, whether it is directly acknowledged as it is in Buddhism or indirectly as it is in Judeo-Christian traditions.
Spiritual Best Practices
Modern spirituality borrows rituals and traditions from every culture, as well as the science of psychology, to create a patchwork of religious, cultural, and secular practices that can have widely varying degrees of success.
From this there needs to evolve a more purely evidence-based spiritual tradition that illuminates the intense beauty of the universe with the inspirational language of mysticism. One that taps the power of our ancestors and practices perfected over countless generations, while providing demonstrable value with its refined rituals based on spiritual and psychological best practices.
Secular Spirituality
Spirituality need not imply the supernatural and superstitious. It can be any pursuit of self-knowledge and mental self-improvement practice. An enlightened secular spirituality can find this knowledge in any source, while maintaining a healthy skepticism for spiritual quackery that runs rampant when science and spirit are separate.
Seeking to learn the science behind ancient practices like yoga, prayer and meditation is a spiritual pursuit. Understanding exactly how rituals affect our brains and bodies doesn't demystify them or negate the placebo effect that is at the root of many health benefits.
While sometimes science will prove some cherished tradition to be ineffective or even counter-productive, it will back with the weight of evidence those that really help make our lives better and allow them to continue to be a part of our traditions when spiritual literalism has become a distant memory.