For the double bass, tenor clef is unnecessary, a bit old fashioned, and should be avoided.
There are two common approaches used in published double bass arrangements:
Use three clefs: bass, tenor and treble.
Use two clefs: bass and treble.
To be clear, both approaches exist in published classical music. There are plenty of arrangements that use tenor clef. There are plenty of classical arrangements that don't use tenor clef and just use bass and treble. For example, I own two copies of Bottesini's Reverie, one uses bass, tenor and treble clefs; the other uses only bass and treble.
Outside of classical music, tenor clef is alien and mustn't be used.
That said, just because both approaches are common, it doesn't mean they are equally as good.
The theoretical advantage of tenor clef is that it reduces the number of ledger lines. In reality, bassists are very comfortable reading ledger lines in bass clef up to an A (3½ ledger lines). At that point you should be using treble clef anyway regardless of if you are using tenor clef or not.
There are two main disadvantages. The first is that the double bass is a true multi-genre instrument. Bassists come to the instrument from many different paths. While all professional classically-trained bassists can read tenor clef fluently, when it comes to amateurs or jazz bassists moonlighting in their local orchestra, they may be uncomfortable reading tenor clef.
The second disadvantage is conceptual and physical. The bass has two registers: normal position and thumb position. When parts use two clefs, the convention is to use bass clef for normal position, and treble clef for thumb position. This communicates useful information to the player because when they see the treble clef they start to think about switching positions. Switching between positions is a large physical arm movement on the double bass!
When tenor clef is used, especially if it's a poor arrangement, the transition between normal position and thumb position may occur within a tenor clef passage. So we have lost that big sign that says "it's time to think about switching". This, for me, is much more important than the ledger lines.
In what ranges should you use bass and treble clefs?
E2 to Db4 is bass clef only zone.
D4 to G4 is the transition zone. Bassists can play these in either position, depending on context. For example, for slower passages, we would usually prefer to exit thumb position and play on the G string because it sounds clearer and more sonorous - bass strings are much thicker than cello strings. For faster passages, this is not possible so we will stay in thumb position and go across to the D or even A string.
G#4 upwards is thumb position zone, and therefore you should use treble clef for this register.
(these are all written pitch, not sounding pitch)
The bass actually has a "secret" third register, which is playing upper harmonics. While the use of this register is usually only found in solo works, the convention is to use treble clef and an 8va for this register.