There seems to be 3 different opinions in the Rishonim:
1. Rambam (הל' ציצית פרק א' הל' ו) writes that 1 out of the 8 strings should be תכלת
2. The Raavad there disagrees and says that 2 of 8 strings should be תכלת
3. Rashi and Tosafos in מונחות ל"ח hold that half of the strings (4) should be תכלת
There seems to be a fundamental dispute. Rashi and Tosafos seem to hold that תכלת and white are equal and together they make up ציצית. The Rambam however, seems to hold that תכלת is used to wrap around the white, there is a fundamental distinction between the 2, white are the strings and תכלת is the wrapper.Therefore, only 1 string is תכלת.
How do we pasken? Here we run into the same issue that I mentioned by Shemitta, we have no mesora of psak. We have a machlokes harishonim but no הכרעה. The Shulchan Aruch, Rama, Magen Avraham, etc. do not discuss the issue because it was not הלכה למעשה. Therefore, we now have to pasken with a 700 year gap.
What makes it worse is that the shitos seem to be exclusionary. If you do like the Rambam (1 string תכלת) then for sure you are not יוצא the din of תכלת according to Rashi and Tosafos as according to them you need 4 strings of תכלת. On the other hand, if you do like Rashi and Tosafos, according to the Rambam you may not be יוצא the din of white strings, because it would seem according to the Rambam you need 7 white strings.
It would be interesting to hear from those who wear תכלת how many strings they have תכלת and why.
Update
It seems that more rishonim and acharonim then I thought dealt with the issue and most pasken like Rashi and Tosafos.
Almost all the חכמי אשכנז (Maharam MiRutenberg, Mordechai, Rid, Semak, etc.) pasken like Rashi and Tosafos as well as both the Rosh and the Tur.
The majority of acharonim (who discuss the issue) also seem to pasken like Rashi and Tosafos.
Taz - או"ח סימן ט' סעיף ב'
בזמן שהיה תכלת מצוותו שיקח שני חוטין וכו': דכתיב גדילים תעשה לך אין גדיל פחות
משנים - גדילים ארבע והיינו שני מינים וסברא שיהיו שני המינים שווים שני חוטים מהאי
מינא ושנים מהאי מינא
Gra - או"ח סימן י"א סעיף י"ב is משמע that he paskens like Tosafos (although some argue that he holds like the Rambam)
Mishna Berura - סימן ט' ס"ק ז paskens like Rashi and Tosafos (half techeles half white):
בזמן הזה - פי' דבזמן שהיה תכלת שהיה צריך להטיל ד' חוטין שני חוטי לבן ושני חוטי
תכלת
The Aruch Hashulchan mentions the machlokes Rashi and the Rambam and is not מכריעה
Shulchan Aruch Harav - הל' ציצית סימן ט' סעיף ד'
י"א שאין ציצית של צמר ופשתים פוטרין בבגדים של שאר מינים אלא א"כ יהיו הציצית צמר
ופשתים יחדיו דהיינו ב' חוטין תכלת וב' חוטין פשתים...
The bottom line is that the majority of the Rishonim amd Acharonim hold like Rashi and Tosafos that there should be 4 strings of Techeles.
Based on this I am very surprised that most people who wear techeles wear only 2 blue strings.
