Read the trust agreement to see how an alternate successor trustee should be appointed. If all named persons decline, they should do so in writing. A last resort, is to go to the court for appointment of a successor trustee. You did not say if this is a "testamentary trust" (a trust within a Will) or a stand alone trust (revocable or irrevocable trust). Hopefully, you will find the answer in the document itself. If you haven't already done so, please consult an estate administration attorney to see if you need to go to probate court or circuit civil court. It can depend on what kind of trust this is. Good luck.
It is very important to read the trust. It should give directions on HOW a trustee resigns - usually in writing and usually with 30 days notice, etc. The Trust may also say HOW to fill a vacancy - for example, all the beneficiaries may be able to vote for a new trustee. However, if all the named trustees are refusing to act, does that imply there is a problem with the trust, or a family dispute? If so, you may want the probate court involved to supervise. Please feel free to contact me if you need more help.
The trust document is normally crafted by the Attorney to provide for this very situation. If ALL of the Successor Trustees resigned (for whatever reason), then a Court of competent jurisdiction can appoint a Trustee. See an Estate Planning lawyer to take this further. See Avvo.Com under Find-A-Lawyer. Good Luck.
Any existing trustee needs to register the trust and seek instruction from the Probate Court. MCL 700.7209 provides instruction as to where the registration may be made. MCL 700.1401 indicates who needs to be served notice. MCL 700.7201, 7202 & 7203 deal with the jurisdiction of the Court to deal with the issues. More than that cannot be determined without reviewing the trust and determining the factual circumstances that you allude to in your question.
As previously stated, the trust should tell you the appropriate individuals who are to be notifed and how they are to be notifed when a trustee resigns. Yet, when in doubt, it is best to ask a lawyer because the failure to give notice to the proper people could end up complicating the matter further. Depending upon your situation, the interested persons may be able to use a nonjudicial settment agreement according to MCL 700.7111(3)(d) to select a new trustee without probate court involvement. It is always possible to go to the probate court and have them approve your settlement agreement, but it is not always necessary. Whether or not this is an option will depend upon the specifics of the trust, the relationship among the interested persons and the possible successor trustees.