Depending upon how you view the Marvel Universe, Cosmic Marvel can come with a host of different views. Most refer to it as the space faring Marvel stories or those that involve a form of cosmic power struggle. With that in mind, I'll try to cover all that can be considered if even loosely.
So first up we have X-23's first contact during her single issue team up with the cosmic force known as the uni-power. The comic Captain Universe/X-23 with the story Double Threat is her first exposure to a cosmic level power in the form of, well, Captain Universe.
No space travels in this issue. The uni-power needed Laura's help to free itself and destroy research done on it by a terrorist organization. Having been fragmented and weaker than normal, it left a piece of itself still in Laura when it moved on.
Next we have the question of Limbo and whether or not it counts as part of the Marvel Cosmic. During the New X-Men: Academy X story Quest for Magik, Laura was pulled into Bellasco's realm. The story began in New X-Men issue 37 and ends with issue 41.
Though often Limbo itself is confused for hell, it is considered a dimensional zone unto itself. Due to it's magical nature though, it is often of much fan debate as to what exactly it entails. Even Mephisto in the 616 Marvel Universe has made bids to control it due to its unique properties. As is considered with most dimensional planes, Limbo has its own laws of nature and physics that do not correspond to any other plane but itself. Whether it's part of the Cosmic Marvel or Occult Marvel(or if indeed those two are one and the same) is open for much debate and exploration as many stories that dive into its nature are never quite clear. Most of them fall upon the perception of the characters visiting or that have been forced to reside there for long durations. Another interesting note, as a dimensional plane it has a different form of time passage in comparison to the standard 616 Marvel Universe. What can be a day on earth can be upwards of 10 years in Limbo.
The next Cosmic level story we would see Laura in, isn't until her volume 3 ongoing comic. During the Chaos Theory story arc, the remains of the uni-power fully make itself known. It reveals itself during an attack by Whirldemons. The fragments of it left behind in Sue Storm, Peter Parker, and Laura activate thanks to Valeria, and send them briefly to the dimension the Whirldemons are sealed into. After rescuing and sending all three home to face the Whirldemon king, it has a brief discussion with Laura about her soul-mark. A symbol that reflects personal choice and growth for a defining moment when Laura decided to choose herself instead of being allowed to be manipulated.
It helps Laura protect the earth and the Richards family from Whirldemons starting in issue 13 with the beginning of the 4 part Chaos Theory Arc. Laura tricks the freed Whirldemon king back to its realm that's oft connected to the Micro-verse, but with the uni-power's help she manages to hold them back, seal them away once again, and escape. The uni-power confesses that Laura is its chosen sword for the future and helps her get back to earth. The Whirldemon king had made a similar remark, stating that the 'soul-mark' that rests in the palm of Laura's hand is a sigil that represents the star sword. Her memory of these events are never made clear if they were left intact as she suggests having dreamlike knowledge of the events after diving into the portal. The memories being somewhat fuzzy after her self-sacrifice. An act she does twice in the arc. The first time to save Sue Richards. The second time to save Sue's daughter Valeria.
Her next story after those events in Chaos Theory is Misadventures in Babysitting. Again co-starring the FF, this story is Laura's first real space adventure as she interacts with one of the Elders of the Universe, Taneleer Tivan, otherwise known as the Collector.
Starting in issue 17 of her volume 3 comic and ending in issue 19, this story arc contains Laura's first space travel, space walk, and interaction with an Elder of the Universe. Just another normal day babysitting Valeria and Franklin for the Richards family. This was also Laura's first interaction with a Dragon like alien of indeterminate origin and species. This story is often hailed as one of the shining moments of the comic run and is often followed by bewilderment as to why this comic was canceled not long after with issue 21.
During issues 32 and 33 of Avengers Academy, Laura has her first real direct exposure to the cosmic entity known as the Phoenix through Emma Frost. While Emma was part of the Phoenix 5, she visited the Academy to forcefully have them dismantle a Sentinel that was in the care of a student.
This story also furthered the rift between Laura and Emma Frost, cementing Laura's opinion of her due to these actions alongside previous multiple attempts of telepathic manipulation or coercion. Laura has long since found an unspecified way to keep Emma out of her head from ever attempting such actions again. This technique was never put to the test with Emma's enhanced Phoenix power level. As that was temporary, we may not see such usages from Emma anytime soon. Though the question could be raised if such unspecified techniques are applicable to the young Jean Grey of the original 5.
The other minor interactions Laura has had with cosmic level beings or threats has been in Avengers Arena. Cammi and Cullen Bloodstone yield two different aspects of cosmic interaction. Cammi having been with the Guardians of the Galaxy allows Laura to meet her first member of their crew, and Bloodstone is another on the list of dimensional plane hopping characters. In Cullen's instance it also brings with him a creature from his previous stay in such a dimension, that of a Glaratrax that he turns into when not in contact with his Bloodstone ring.
Another character of note is Darkhawk via Avengers Arena as his armor and erased internal AI are part of the cosmic order called The Fraternity of the Raptors. It has been mentioned before in his appearances in Nova and the War of Kings story arc that other amulets are winking into existence all over the universe, but that story has yet to be fully told or elaborated on. Chris Powell (the original earth Darkhawk) has yet to fully interact with Laura during their stay on Arcade's Island, but due to his concept and more recently created premise as revealed during War of Kings, it should be acknowledged that he does indeed count as part of Cosmic Marvel as do all the amulets from the Fraternity that are showing up across the universe now.
As of this writing at least, these are X-23's cosmic adventures or stories that relate to such interactions of dimensional planes and outer space or what can be considered Marvel Cosmic level entities. I didn't include Circle of Four as it's currently pretty solidly within the confines of Occult Marvel. Also missing from this list are X-23's Marvel Team-Up appearances.
With the the Trial of Jean Grey slowly approaching that concerns what may be Laura's new team for her to look after, it'll be interesting to see if it can be added to this list. Laura is a character that has done ample amounts of research on everything whenever she can. Having read the X-Men database while with the New X-Men, and potentially reading the Avengers database after her stay with Avengers Academy too, it could prove interesting what insights she may have and how she adapts to such an adventure as well as the characters she could meet.