There's a lot to choose from, but each Overdrive has it's own voice and style of grit. Each one will react to your bass and amp differently, so don't expect every Overdrive to work for you in the same way that it does for other bassists. Your gear is different and that will cause an Overdrive to react differently too. So here's my take on dirt.
The MXR bass overdrive is an OK overdrive, it's not particularly dynamic or transparent...but it plays really well with other XMR pedals. Pop it between an MXR Bass Octave Deluxe and MXR Envelope Filter and it really shines.
I have three fuzz boxes, so lets get that out of the way too. Fuzz isn't overdrive...but some fuzzes can overdrive and some overdrives can fuzz...but they never quite sound right in my opinion. So I like my fuzzes to be very heavy and have a certain sound mashes my bass into. So I use a Woolly Mammoth, a Swollen Pickle and a MXR El Grande. All of which I really like as stand alone sounds. But they also shine with strong ambient reverbs places after. If you are going to do "whale song" sounds...I do it with a heavy fuzz and my base tone.
In overdrive land...your original request...
My favorite three Overdrives are a Wampler Euphoria/Ecstasy, a V4 OCD with various mods and a Fairfield Barbershop. The Wampler is a very smooth unit but offers three clipping modes and has a bass / treble controls which work well on bass. The grit can be very light, smooth or quite aggressive and this pedal can also be used as a clean boost or a bass and treble cut /boost EQ. It's very versatile and if I was to choose only one OD...it would probably be this one. I place this in front of my Barbershop so I can clean boost it and get more grit out of it. It also cascades with other OD's well. Which leads me to the Barbershop...a very transparent OD, which seems to affect a narrow tone band (mids?) with its grit. It's only got one grit sound and all of my basses still sound like my basses through this pedal, but with added grit. It's quite a light overdrive, but can be pushed (using another pedal) into quite heavy overdrive /fuzz with a boost. It gets quite gnarly and has a unique sound in that zone...but it's not for everyone. As a light overdrive, it's amazing...it can do tubey pre-break up...it can so just breaking up...it can do mild overdrive...it's very touch sensitive and a lot of fun. The sag control brings in a level of compression, which is better served with a full compressor...but it does add another "sound" to the pedal. My last OD pedal, which is placed first in my chain, is my mk4 OCD. This is a clone supplied by Fuzz dog and it's got the more bass switches and 18/9v option too. I REALLY like this pedal and it's very versatile. It can go from completely clean to very heavy overdrive. It's got a great grit structure and it seems to work well with a lot of basses. It will add an OCD stamp on your tone, but it's ok because it really does sound like a heavy tube amp being punished with some high gain (every type of tube amp had it's own sound and it's true of OD's too).
Other worthy mentions are the Joyo American Sound...it's an amazing unit, the Blueberry (in all of it's guises) is excellent too. Dam Ezekiel is uber-versatile and again excellent. The Timmy is awesome too, but the tone controls are cut only and the frequencies need tweaking to make them right on bass. The Sweet Honey Overdrive is a little bass light but makes a really nice sound on most of my basses.