David Reilly and colleagues (Lancet 1986; ii: 881-886) compared a homeopathic pollen remedy (30C) to placebo in hay fever in 144 human subjects. They showed a small but statistically significant worsening of symptoms during the first two weeks of treatment but a small but significant improvement in the two weeks after completion of the treatment that was associated with a reduction in the amount of antihistamine needed to control their symptoms. A summary of the published critiques: 

Despite the huge potential impact, the study has never been repeated. This is critical in the study of medicine. Clinical trials are notoriously subject to unnoticed influences and effects. All medical observations must be repeatable in a scientifically verifiable way. (Note: An additional study, in which allergic asthma was the topic, was performed subsequently by Reilly, et al. (Lancet 1994 Dec 10; 344(8937):1601-6). The results of this very small (28 patients) were similarly positive, but suffer from the same deficiencies as the other study and do not constitute a replication of earlier work. 

The assessment of the response was subjective. 

Only 67% of the patients who started the study finished it. This is a low number of completions for a 5 week study. 

The chemical composition of the treatment and the placebo was not assessed. 

Hay fever exhibits a large placebo response and has a waxing and waning character that makes it hard to study clinically. 

Migraine study: 

Brigo et al (Berlin J. Res Homeopathy 1991; 1: 98-106) reported that a homeopathic remedy performed significantly and dramatically better than placebo in a randomized study

Subsequent studies by Whitmarsh and colleagues at Charing Cross Hospital showed no effect of such treatments compared to placebo (Cephalalgia. 1997 Aug; 17(5):600-4; J Altern Complement Med 1997 Winter; 3(4):307-10). Thus the study of Brigo, et al was not reproducible, and homeopathic treatment was no better than placebo. 

It should be noted that in the case of these migraine trials a number of patients did undergo dramatic responses; in the Whitmarsh study, the responses were evenly distributed between the placebo and treatment group, demonstrating the robustness of the placebo response in clinical trials

Benveniste Affair: 

Benveniste and colleagues published a paper in Nature (Nature 1988; 333:316-318) that appeared to demonstrate that homeopathically prepared antiserum for IgE could cause human basophil degranulation. This appeared to be clear evidence of the efficacy of a homeopathic remedy, since no placebo response was possible in isolated cells.

A team of scientists, including the noted skeptic James Randi, visited the Benveniste lab and concluded that the experimental methods employed were flawed (Nature 1988; 333: 287-290). 

A subsequent study by Hirst et al., duplicated the conditions under which Benveniste reported positive results and could find no effect of the treatment on degranulation (Nature 1993; 366: 525-527.