I have a somewhat unique approach to playing bass, much more aggressive lines and sometimes play what would be considered a lead line if a guitar was playing it (or other lead instrument). With that said, my current setup is as follows:
Tuner pedal- Obviously used for tuning but can be very helpful to act as a mute. This is especially helpful if you need to switch instruments between songs. I also like it so that I can make sure I am being quiet in between songs while the singer is talking (no one likes a noodler getting in the way of addressing the audience).
EQ pedal - My main use of the EQ pedal is to be able to switch from my normal, thicker bass sound to a punchier sound with more high end. This has two particular uses for me. One, I like to switch between slap and fingerstyle within a single song, so I will use the EQ pedal to get the slap sound I'm looking for (less muddy with more string noise). Two, when I play lead parts or solos I use the pedal. Adding more high end to your sound will allow it to cut through the mix a little more so that your lead parts aren't competing with the other instruments that usually take up the forefront. EQ pedals also typically have an overall gain/volume, often referred to as "compensation gain", as in it can be used to compensate for any change in volume that results from the EQ changes, allowing you to have an output volume that matches the volume of your instrument without the pedal. This gain compensation can also be used as a volume boost, which can especially be helpful for lead/solo parts.
Bass Chorus- I set the chorus up to have more of a subtle effect, ideally so that no one would really notice it was there. It allows me to gain more presence in the mix but is more suitable for me when adding presence to the fingerstyle sound than the EQ. It makes my sound thicker without the extra volume and allows me to pluck the strings relatively softly (which generally has a warmer, thicker tone) while still getting the presence. You can also use the Chorus pedal to apply as an effect that will thoroughly change the sound. With more drastic settings than I use, you can get a tone and effect that is anywhere from a noticeable chorus effect like guitarists often use all the way to a sound that feels out of tune or actually oscillates the frequency in a way that doesn't really function for a normal bass part.
Delay Pedal- I use this more as a reverb unit by setting the delay time very short (ideally under 30 milliseconds, which is the amount of time that the human brain can perceive two sounds as separate). I also use it as a sort of sustain pedal. The reverb type application is really only used when I play certain lead parts, particularly in one song I play where I use artificial harmonics in the intro, which sounds kind of like a keyboard. As a sustain effect I really only use it in one or two songs. It allows me to get big whole notes or double whole notes that don't decay before the note is really finished, keeping it more present in the mix. WARNING: Most bass players do not use any sort of reverb. This is because it can make things very, very muddy in the overall mix, especially if the very low frequencies are not filtered out of the effect sound. I have very specific uses for this pedal, which often do not apply to other bass players' approaches.
Envelope/Bass Synth Pedal- This is a pedal I like to use but has not had as much play for me in quite a while. One large reason why it hasn't been used as much is that it is a volume sensitive pedal, ie, the input volume controls the reaction of the effect, and I play an active bass, making the volume too loud to get the full range of sensitivity. This pedal can create sounds like Bootsy Collins or Les Claypool have used. It can also be used to create a synth sound that can be good for replicating the sound from certain songs, especially hiphop, modern pop, or songs from the 80s, that use actual synthesizers. I have been using it most recently to imitate the synth sound in the intro of "Beat It" by Michael Jackson, which my band has been covering.
I also have a looping pedal but that has only really been particularly effective for practicing on my own and I had considered using it to perform solo but never actually followed through. Lopping pedals are a lot of fun but I have only had one instance of a group I played in where it was useful. We were a rather unique hiphop influenced free improv group. Our vocalist would beatbox and I would play a bass line and that would be the core of our song and we could then add layers and have a solid backing track for rapping on top or lead parts/solos.
I have been considering getting some sort of distortion pedal as well. I have used an overdrive and it had a great sound but I've also heard plenty of more distorted sounds that have worked well. This is usually most applicable for heavier rock styles of music but can certainly have a use in other settings as well.
As others have mentioned, compression is one of the most commonly used bass effects. This is really used to modify the volume output for more consistency than to actually change the sound, though many compressors have a tone or color to them that can add a lot to a player's sound. When I hear it used for tone, it is really just on all the time, so their tone at all times is based on this. I haven't really found a compressor that I like in pedal form, which is the only reason I don't have one. A lot of bass players actually have a rack-mount compressor instead of a pedal that runs in line before the signal reaches the amplifier.
Depending on what style of music you are playing and how you apply them, it is usually best as a bass player to use effects sparingly. Some effects will take away from the role that the bass is "supposed" to fill or can take up space in the overall mix. So when I use pedals, I prefer to have either a very subtle effect that modifies the listener's perception of my sound or presence, or try to use them in specific places for a specific reason. Players like Bootsy and Claypool use a lot of effects very frequently (seemingly all the time) and I thoroughly enjoy their application, however, this does not translate to all genres by any means. You will also likely find that many players that you might collaborate with are not looking for a bass sound with that much presence; they are likely looking more for the traditional function out of a bass player. In such a setting, a good use of an effect might be on just one song, in just one section, most likely the bridge. This can allow that one or two things you use it for to be fun and different, bringing something new to a potentially boring section. Occasional use can also allow the audience to enjoy it more, maybe even waiting for that section because it is "super cool!". If you use the effects all the time, they become a part of your sound and your style, such as Bootsy and Claypool, which can be good or bad, depending on what you or the band you are playing with are looking for.