Let me preface things here: Besides being a tubaist, I am a serious conductor who for the past 35+ years has worked with select groups of mostly professional musicians. From many years of experience I have a very good take when it comes to listening to new players. It usually boils down to "do they have the skill sets that I need for my groups?" There are many different types of players, from classical, jazz, broadway, chamber, solo etc, etc, etc. and various schools of playing within all those subcultures. It becomes a matter of matching the player with the appropriate skills in order to get the job done.
Now: It seems like most of the issues have been address and rather well. The important point for me - when I audition or inquire is the level and agenda of the band. Yes, some are rather competitive and have a preference for modern or traditional style of repertoire. I can think of a handful here in the greater Minneapolis area.
A wise conductor will be looking for certain skill sets, which will probably be obviously demonstrated in the first few moments of the audition. Rhythmic accuracy, technique, tone and intonation, general musicality (usually displayed in more legato playing) and a certain confidence or aggressiveness in your approach. Some might ask you to sightread a passage to get a feeling for how well you actually read. I suppose in cases that are not so competitive,the 'gauntlet' may no be so rigorous. But if you want in, you need to show them that you are the best candidate. Prepare, prepare & prepare. Then, relax, relax & relax, and then have fun playing the audition. Being tense and stressed never helped anyone. Talk to whoever is in charge of the audition well ahead of time and try to get a fix on what to expect and how much time will be allotted to you. Also try to be very well warmed up before hand and play into the room.
Etudes and solos, carefully chosen to showcase one's skills are a wise decision.
For tuba etudes I tend to like Blazevich Vol 1 & Bordogni. There is so much in them to demonstrate mature playing. Tyrrell is good, but they tend to be mostly receptive patterns.
Pick two that are contrasting in style and technicality.
For 'solos': Well that is a pretty open issue. So many to choose from - but they should be extremely well prepared. Yes. it could be (from my list of favorites) Air and Bouree, the Capuzzi Rondo, Beelzebub, The Carioca, Dave Uber's Jazz Rhapsody, or a movement from the Vaughan Williams.
What I have found with 'excerpts' is that most (not all) modern tuba (and even euphonium) band parts are no where near the level they were a couple of generations back. Granted much of the "meat and potatoes" were the transcriptions from orchestral music and marches. They required technique and strong chops.
I'm thinking of:
Russlan and Ludmilla
Marche Slave
Entry of the Gladiators
Die Meistersinger Prelude
Yes: the Chaconne from the Holst 1st Suite (for kicks prepare it an octave lower as well)
Then there are the "dog fights" from circus marches and galops.
Circus Bee
B & B's Favorite
Storming of El Caney
Storm and Sunshine
Sousa: Free Lance March.
Non of these are for the "faint of heart" (especially 'at tempo') but they have appeared on major service band auditions and would be suitable for 'competitive' community bands. The other issue is where to find these and other parts. Most are at the Chatfield Band Music Lending Library. Here is the link:
http://chatfieldband.lib.mn.us" target="_blank
Good luck. ... and remember - music is to be enjoyed.
A rather wise conductor, Keith House, from Central Methodist once said: "relax, it all depends on you."
There are only so many decent groups that utilize tuba 
