Assuming you're referring to the immediate aftermath of 476 when Odoacer deposes Romulus Augustulus, then there's two notable rump states worth mentioning, though neither would last for an extended period of time.
First is the province of Dalmatia which was governed by the deposed Western Emperor Julius Nepos. After Odoacer deposes Augustulus, he asked the eastern emperor Zeno to sanction his rule of Italy and grant him the title of patrician. Zeno told him that he should ask Nepos who was the rightful emperor of the western empire - but also technically recognized Odoacer as ruler of Italy and as a patrician which was good enough for Odoacer. Odoacer paid lip service to Nepos - barely. He minted coins in Nepos' name, but never allowed Nepos to actually return to power. Nepos was assassinated in 480 in a plot some believe was backed by Odoacer. Either way, Odoacer "avenged" Nepos and took occupation of Dalmatia for himself with the blessing of Zeno.
The other notable rump state was known as the Kingdom of the Soissons and was located in northern Gaul (modern day France). In 458, Emperor Majorian names Aegidius the magister militum of Gaul - which essentially meant he was the head of every Roman army in Gaul. Try as he might to beat back the Germanic tribes flooding into Gaul, he was unable to hold the territory in Southern Gaul and his territory in Northern Gaul was cut off from the rest of the empire. Aegidius dies in 465 and his son Syagrius takes over. Syagrius uses the title dux which was the title of Roman provincial commanders (and the base for the latter feudal title Duke). However many of the neighboring Germans began referring to him as king of the Romans, however Syagrius himself is not documented as referring to himself as anything more than governing a Roman province. In 476, Syagrius sends word to Zeno rejecting Odoacer's rule and says his armies are ready to invade Italy, but Zeno doesn't budge and that's the last time the eastern empire noted it heard from Syagrius. In 486-487, Syagrius is defeated by the Frankish king Clovis and executed.
Penny MacGeorge. Late Roman Warlords
To build upon what u/Manofthedecade started, the collapse of the west was a long process, and a complex one at that. Over the last 25 years or so, starting around 450, semi-independent or independent "warlords" start to pop up. I'm going to divide this response into the more well known, "large-scale" ones, and then small-scale breakdown like we see in Noricum (and possibly Britain).
The most notable "warlords" of this period are Aegidius and Syagrius in the so-called "Domain of Soissons" (which never existed, a topic I'll get to in a bit), and Marcellinus and Julius Nepos out of Dalmatia. But there were a lot of smaller ones too: Hubaldus of Rouen, Arbogast of Trier, and the Comites Blaesensis and Castrodunensis, as well as Paul of Angers.
The Domain of Soissons is a complex topic. To understand it we need to look at it in a greater context, namely the relationship between the Romans and the Franks of North Gaul. Aegidius is often simply stated as magister militum per gallias and it's left at that, but that doesn't actually seem to be the case. In fact, the magister militum per Gallias from 452-455 was Agrippinus, before he was probably replaced by a Gothic candidate under Avitus, possibly Frederic. We know little about Aegidius' origins, other than an off quotation from Priscus that says he served with Majorian under Flavius Aetius. Majorian served in north Gaul and (at least according to my own theory) was likely appointed magister militum per Gallias after 439 and served until 450 or 451. It's unknown what command Aegidius held, and his association with Soissons actually doesn't say much. He could have been dux Belgicae secundae, in charge of the Sambre-Meuse limes, which would explain his close relationship with the Frankish foederati. But he also could have been a civil administrator for the region. His association with Soissons (Suessiones or Noviodunum) would suggest he may have actually come out of the logistical branch of the bureaucracy, possibly originally being in charge of the Roman fabrica (arms factory) there.
Aegidius, though, doesn't seem to have actually "split off" from the empire at any point. He acknowledged Marcian and then Leo I's rule, and seems to have considered Avitus something akin to an usurper. He didn't split away from Avitus, but didn't acknowledge his legitimacy, and well for him as Avitus was quickly defeated by Ricimer and Majorian, as the Italics didn't want a Gaul or a man promoted by the Aquitanian Goths to the throne. When Majorian is made emperor and Leo acknowledges it, Aegidius does as well, and Majorian makes him magister militum per Gallias in exchange. In reality, this may not so much be a reflection of Aegidius' command ability as Majorian's political sensibility, since he needed the support of the Gallic aristocrats. When his forces joined with Majorian's in Majorian's reconquest, he seems to have immediately accepted Nepotianus as the supreme commander in the campaign. But our sources for Majorian's reign are sparse, so we don't really know the specifics of this campaign. We can only guess.
Upon Majorian's assassination by Ricimer, Aegidius and Leo I (as well as Marcellinus, who I'll get to) both refused to acknowledge the accession of Libius Severus. Again, Aegidius technically wasn't splitting away from the empire, he just wasn't acknowledging basically what he saw as another usurper appointed by a king broker, Ricimer. Ricimer had Aegidius demoted from his position and reappointed Agrippinus (who may have himself been from Gaul). Agrippinus enlisted the help of the Aquitanian Goths and apparently there was a long standoff or skirmishing period before the Goths finally mounted a large-scale attack over the Loire in 463, targeting the stronghold of Aurelianum. This is where another major figure truly enters the scene: Childeric. Childeric seems to have succeeded as "king" of the Frankish foederati in 457/458, and may have possibly been granted the title of dux Belgicae secundae (the letter of Remigus is somewhat suspect, see Halsall's work on that). Franks, Alamanni, and Alans probably composed the majority of Aegidius' forces. Thus when Aegidius died in 464/465, Childeric became the preeminent power in the region. There's much talk of how Syagrius "held out" but in reality Syagrius was probably still a boy or a teen when Aegidius died, and the primary sources indicate Childeric and one "Adovacrius" (possibly Odoacer, king of the Sciri, who may have still been operating in Gaul since the events of Attila's invasion in 451), along with another local warlord, the comes (probably comes tractus aremoriciani) Paulus of Adecavus (Angers). This time, Paul, Childeric, and Odoacer beat the Aquitanian Goths as they had been in 463, but Childeric then turned on Paul and took Angers. Odoacer turned on Childeric, but Childeric defeated him as well. Barring the comites blaesensis et castrodunensis, as well as Armorica (Brittany), this left Childeric in control of everything North of the Loire west of Arbogast in Trier (who may not have been Roman, but Frankish or Alamannic anyways, judging by his name).
So where does this put Syagrius and his "Domain of Soissons?" Well the preeminent theory of both Edward James and Ian Wood is that there was no "Domain of Soissons." Instead, Syagrius became a vassal of Childeric upon Aegidius' death. He was too young to rule on his own, and most of his forces would have been loyal to Childeric anyways. The events of 486 to 487 actually read as an internal rebellion in Gregory of Tours, not a conflict between two separate entities, when Clovis defeats Syagrius and gives Soissons to his cousin Ragnachar.
(1/3): Next up Marcellinus.