There is a distinction to be drawn between, on the one hand, the occurrence of evolution (i.e., the fact that species change their forms over time), and its general pattern, and, on the other hand, the mechanism by which this occurs, such as natural selection. There is not much room for debate about the occurrence or its pattern, but a tiny bit of wiggle room over the mechanism.
Lamark actually had a different view of both of these issues from those held by Darwin and modern biologists. As most people are aware, Darwin (and biologists today) understood evolution as a branching process, a process of differentiation of species, with one species sometimes giving rise to another distinct one, while the original species continued to live on. Also species might sometimes become extinct. It is usually assumed that there was one single original type of living thing on Earth (I think Darwin himself allowed for the possibility that there might have been a few separate origins, but certainly not nearly as many as there are species today) and everything else branched off from there. This gives rise to the familiar sorts of branching tree diagrams of evolution.
Lamark, however, did not believe in branching evolution. As he saw it, each species had a separate origin, but it would also evolve over time so that the modern form of some type of creature might be very different from its earlier ancestors. Thus a Lamarkian diagram of evolution would show many parallel evolving lineages, rather than a branching tree. Also, Lamark thought if a modern day species had a relatively simple structure, that meant that its lineage had originated relatively recently and it had not yet had enough time to evolve into a more complex form. Everything started off simple (unicellular I guess) and got more complex over time. As I understand it, a large part of the motivation for the whole system was than nothing should actually ever go extinct - fossils of things that are no longer found alive were actually just earlier forms of things that had now evolved into something else - because if a species actually ever went extinct that would mean that God had screwed up, which was religiously unacceptable. (The worries that devout Christians had about biology in Lamark’s, or, come to that, Darwin’s, time had little in common with the Bible idiolatry that developed in 20th century America. Modern creationism is rooted in ignorance about Christianity quite as much as ignorance about biology.)
As for mechanisms of evolution, so far as I am aware the only serious candidates are Darwinian natural selection (as amended by modern genetic knowledge), Lamarkian inheritance of acquired characteristics (whereby improvements an individual organism manages to make in itself over its lifetime get passed on to its offspring, who then make further improvements, and so on), and random genetic drift.
I do not think Lamarkian parallel lines of evolution would find any supporters anywhere these days. On the one hand it has no appeal for the Biblical literalists (even if you stipulate that the original ancestor in each parallel lineage was created directly by God), but, more importantly, it just does not have the explanatory power of the Darwinian branching model, which explains all sorts of facts, such as the pattern of geographical distribution of species and the hierarchical pattern of similarities and differences between species, that the Lamarkian theory has no more to say about than crude creationism does. (Some of theses facts, such as the geographical distribution of species in isolated environments like remote islands, also make no sense from a creationist perspective. If God did it that way he must be nuts! These were the facts that persuaded Darwin himself that evolution was real, but not Lamarkian, well before he came up with natural selection as a mechanism to drive it.)
It is a bit different when we come to the mechanisms, though. Random genetic drift undoubtedly happens, and has real effects, although I believe the scientific consensus is that they are mostly swamped by the much larger effects of natural selection.
Darwin himself was quite prepared to believe that the Lamarkian mechanism might play a significant role in evolution (increasing so in later editions of The Origin of Species), and there is certainly nothing inherently unscientific about the idea. The main problem with it is that given our modern understanding of how inheritance works (DNA and all that), there seems to be no mechanism whereby acquired characteristics could be passed on to offspring, as Lamark’s theory requires. It is not inconceivable, however, that such a mechanism might be discovered. If it was, it might turn out that the Lamarkian mechanism might play a minor role in driving (Darwin style, branching tree) evolution. It is gong to be at most a very minor role, though, because it is clear that natural selection, which certainly occurs, together with a little bit of genetic drift, is plenty enough to account for most (perhaps all) of the facts.
