Miura VS by La Sportiva — I've used this climbing shoe since the first season I ever rock climbed. It is an all around machine, good at slab, vert, and overhung climbing. The shoe is very stiff, and supports the foot for long pitches. It is incredibly precise and is the single best edging shoe that I have ever used. The velcro-closure system allows for a tight-fit thanks to the 3rd strap, but allows the shoes to be popped on and off between pitches. This shoe is a phenomenal all-arounder, but may not fit more "narrow" foot shapes. Additionally, the shoe is stiff, so those looking for a soft and sensitive shoe may be turned off by its aggressive support.
Miura by La Sportiva — There is a reason that this shoe is still selling 30 years after it was released. If I had to only choose 1 pair of climbing shoes for the rest of my life, it would be the Miura lace. This shoe is the ultimate quiver of 1 climbing shoe. It is quite stiff (8/10 stiffness), but still soft enough to climb smeary granite and overhangs. The lace closure system allows for a great fit, and so is comfortable in hand cracks.
The stiffness of the shoe allows it to climb miles of techy stone and still maintain power through the feet. It is slightly downturned, but loses a lot of its downturn after a few pitches, as it lacks the P3 Power platform. This is actually great in my opinion, as the shoe smears better due to its flat shape, and is more comfortable and neutral. The shoe is quite assymetric, and has a pointy toe, allowing it to claw into small pockets and features.
Although many shoes are better at specific terrain (miura VS for tiny edges, Solution comp for giant roofs, testarossa for steeper edging), this shoe will do everything almost as good. If you have to buy one shoe, let it be this one.
One last note, this shoe is QUITE different from the Miura VS. For starters, the Miura VS is stiffer, has a velcro closure, and maintains its downturn shape due to the P3 power platform. Additionally, the Miura Lace version is slightly narrower, and seems to break in a bit faster for me. Although they are both phenomenal shoes, the Lace version is a more all-rounder, while the velcro version is better at tiny edges and sportier terrain.
Solution Comp by La Sportiva — If you are climbing steep walls, this is the shoe for you. I have used this shoe on steep walls in Kalymnos, Red Rock, the VC, and the Red. These shoes are stellar for steep, smeary terrain, although they edge better than any soft shoe that I have ever used. The heel on these shoes is also stellar, and fits most peoples feet much better than the original solution heel.
I don't know how this shoe maintains such good precision and edging, while also being so incredibly soft and sensitive. This is the ultimate modern bouldering/sport climbing shoe.
TC Pro by La Sportiva — The absolute standard for hard multipitch, hard trad, and big wall rock climbing.
These shoes are very stiff, incredibly comfortable, and climb cracks better than any shoe on the market. The raised ankle protection keeps the feet and ankles protected in heinous wide cracks, and the stiff XS edge sole keeps your feet feeling good 10 pitches off the deck. I size these shoes about 0.5 sizes above my sport size, which is the perfect mix of comfort and performance.
They eat hard granite and cracks for lunch. Don't buy these shoes for the gym. They are way too stiff and expensive to be using on plastic.
Mantra by La Sportiva — My favorite gym climbing shoe.
These are fantastic for training, they are light, sensitive, and have a "playful" feel. The no-edge tech works great for indoor climbing, smears and edges well on plastic (and softer rock types like sandstone).
The shoe is incredibly soft (the softest shoe that la sportiva makes, along with the Ondra Comp), which makes it phenomenal for steep bouldering.
I love the slipper-fit, as it makes the shoe lighter and more comfortable for gym/training sessions. It is easy to pop them on and off.
These shoes are just pure fun to climb in. They are soft, light, and sensitive. Seriously feels like a climbing sock!
Solution by La Sportiva — This is one of the most popular shoes on the market.
This shoe was made 20+ years ago, and is still found on the feet of climbers on the hardest routes in the world (V17, 5.15d, 5.14 trad). It is good at every type of rock and style, but where I think it excels is hard bouldering and sport climbing.
This is my go-to bouldering shoe. It is stiff enough to edge really well, but also soft enough to smear on tiny features. It is very sensitive, and a nice toe-patch is great for toe hooks.
The heel is not for everyone, and honestly, it doesn't fit my foot very well. However, It is a hard, "knobby" fit, which makes it really great for hard heel-toe cams or painful, sharp heel hooks. The heel is pretty uncomfortable on my foot (for some it is great!), which is why I only use it bouldering, as I can slip the shoes on and off after each attempt. For a hard, steep sport climbing shoe, I prefer the newer Solution comp, as the heel fits my foot better.