Current discoveries include 18 from Brian Trial, 9922 from Jim Williams, 11324 from Milla and Anderson, 62 from Stephen Johnson and 2 from Ian Gambini.
It is estimated there are close to a million (approximately 990,000) SPSSs in this order.
UPDATE! Jim Williams completed the enumeration of SPSSs of order 34 in April 2016. There are 990981 SPSSs in this order. The number of new discoveries by Jim is 969653, bringing his total discoveries in this order to 979575.
Listings; Bouwkampcodes spsso34.bkp.zip and tablecodes spsso34.txt.zip and postscript spsso34.ps.zip.
The full listings of this order, including postscript graphics of the dissections can be downloaded from the links above.