Last year I purchased and planted an oriental dogwood but it did not leaf this year. In May it started sending suckers from the base of the trunk with healthy shoots and leaves. Are any of these shoots viable to grow a new tree? Do I cut all out but one to allow to grow and do I cut the trunk down to the location where the shoots are forming? Lastly, do I give up on the and dig it out totally?

Same as any other bush/tree which develops suckers - you cut them off as close to the trunk/root as you can and leave the main trunk/roots alone.