There has been one notable mission that went vertically out of the plane of the solar system. That's the Ulysses spacecraft which did a flyby of Jupiter to get into an orbit to study the Sun's poles from above and below.
However this also illustrates the problem. Because the Earth orbits in the plane of the solar system, in the same direction as all the other planets - it requires a huge amount of delta v to do an orbit like this. Ulysees was able to achieve this with a close fly by of Jupiter.
Add to that, that there isn't much else, other than the poles of the sun, of interest out of the plane of the ecliptic, at least close to the sun, and you can see why we haven't done this often.
We do have the roughly spherical Oort cloud of comets orbiting beyond Pluto - but these are far away so unlikely to be visited soon - and probably there are many members of it that orbit close to the ecliptic also.
Voyager 1 is another spacecraft that went almost vertically out of our solar system In this case, wasn't by design, but as a biproduct of the decision to do a close flyby of Titan. Because of its thick atmosphere, the only solid moon or planet in our solar system with a thick atmosphere other than Earth itself they saw it as a prime target to study.
They could have gone on to explore Uranus and Neptune like Voyager 2. They could also have done a flyby to send Voyager 1 on to visit Pluto, but instead decided to do a close flyby of Titan. This seemed a reasonable trade off to the planners - they had no way to know that Titan, interesting though it undoubtedly is, would turn out to be completely shrouded in haze, with no surface features visible to Voyager 1.
As a result, Voyager 1 left our solar system almost vertically, like this:
Taking this image as it left the disk of the solar system
Crescent Saturn as seen from Voyager 1 four days after its flyby
Other ones that left the solar system well out of the plane of the solar system include Voyager 2, it did this to fly by Triton which has an inclined orbit
As it left the plane of the solar system it took photos of Neptune's rings like this one
Pioneer 10 did a flyby of Jupiter which deflected it out of the solar system, then back into it again for the Saturn flyby, after which it went into a solar escape trajectory.
This is an image it took of Jupiter thirteen and a half hours after its polar flyby when it left the plane of the solar system for the first time.
On a smaller scale, locally - the Russians because they launch from the North of Russia, well away from the equator, typically send spacecraft into orbits that go way above and below the equator. The ISS also is in an orbit inclined at a similar angle to the equator, making it easy to get to it with the Soyuz spacecraft.
The Russians especially have launched many satellites into steeply inclined orbits like that, because it is the easiest type of orbit for them to launch into and because the orbits fly over Russia. In particular they have often used Molniya orbits - which are sort of like geostationary orbits - but the satellite is in a very elliptical orbit and spends a lot of time each day high up above the Northern hemisphere looking down on Russia etc - then does a close fly by of the Southern hemisphere for a few hours, swings around and then goes back up to its high vantage point above the Northern hemisphere.
The Hubble space telescope orbit is also quite inclined, not as much as the ISS
There are also, satellites in polar orbits, or close to it, for instance, inclined at an angle of 8 degrees. Because of the bulge of the Earth, a satellite at an orbit inclined at 8 degrees to the vertical precesses at just the right rate so it always has the same orientation to the sun - called sun synchronous, orbit. Useful for instance for studying the aurora.
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