Several opinions claim to have rediscovered Techeiles. This question refers to Ptil Techeiles and Radnyzer. What are their opinions as to the Techeiles, and are there any others?
Radzyner techeiles is made from the "ink" that is secreted from the Sepia Officinalis, i.e. the common cuttlefish. The Radzyner Rebbe, R. Gershon Chanoch Henoch Leiner was the first person in modern times to try to "rediscover techeiles". He wrote a short book called Sefunei Temunei Chol in which he outlines 10 criteria for identifying the creature that would produce techeiles. In the last paragraph he says that he was able to obtain the species that fits all the criteria, but further research with live samples is necessary to actually produce techeiles.
Then he wrote a second book called Pesil Techeiles in which he describes how he went to the aquarium in Naples, Italy and saw the creature and learned everything about it and saw that it was indeed the creature that produces techeiles. In this book he explains how the creature fits all the criteria he laid out in the previous book.
Petil Tekhelet Institute
Eventually, via the research of R. Yitzchak Isaac Herzog, it was discovered that the techeiles produced from the cuttlefish was simply Prussian Blue. That means that the cuttlefish played no role in the dyeing process; it simply provided the organic compound which, when certain chemicals were applied, became Prussian Blue, a synthetic dye. Most people therefore rejected this techeiles, as the same synthetic dye could be produced from any organic compound.
In his research, R. Herzog suggested other candidates. The one used by the Petil Tekhelet Institute (and probably everyone else besides a few Chasidic groups who stick with the Radzyner techeiles) is the Hexaplex Trunculus, more commonly called the Murex Trunculus.
R. Herzog had several issues with this candidate, most notably that no one could actually produce a blue dye from it.
After R. Herzog's death Otto Elsner discovered that the dye could be turned blue by leaving it out in the sun. Since then they have been using this dye as techeiles. Hundreds of articles have been written discussing how the Hexaplex Trunculus, and the dye it produces, fit or don't fit the necessary criteria. The best starting point is probably the series of articles in The Journal of Halacha and Contemporary Society by Dr. Mendel Singer an Dr. Baruch Sterman, which can be read here (Singer argued against the Hexaplex Trunculus, and Sterman argued in favor of it.)
The Janthina was another suggestion by R. Herzog, and it may have fit some of the criteria better than the other candidates but no one was able to produce a blue dye with it. I believe Dr. Saul Kaplan is convinced this is the correct creature, and even claims to be able to produce dye from it, but as far as I know no one has ever been able to publicly produce techeiles from it.
I think this covers the basic views, albeit in a somewhat simplistic manner.