From a pure legal standpoint, trust property is owned by the trustee. From a tax standpoint, if this is a revocable trust, the owner for tax purposes is the person who transferred assets into the trust. If the asset is community property, then technically each spouse owns half the property, and each spouse owns half the asset for trust purposes.
If this is an irrevocable trust, then the trust is usually a separate taxpayer and files Form 1041.