We have already seen that safety of vaccines is tested without a real placebo, but only in the comparison to other vaccine, or in comparison with some other toxic compound. But this is not all. There are a few more problems with testing vaccines’ safety.
First of all, nearly all of trials are done exclusively on healthy children, which does not stop manufacturers, FDA and CDC to recommend the vaccine to not so healthy children, premature babies, and not only children, and also children of a younger age.
Secondly, nearly all clinical trials of safety are looking only for the short-term effects. Usually, they last from several days to several weeks. There are rarely trials that last several months. All of the side effects which happen after this period a-priory can not be related to the inoculation.
Thirdly, even if there are serious adverse effects occurring during the trial, scientists can decide that those adverse effects, or even death, are not related to the inoculation and can cross them out without taking into account.
Fourthly, research is usually done on relatively small groups of children. For example, Hepatitis B vaccine was tested on 147 newborns. Usually, vaccines are tested on children from third world countries, which halves the costs for clinical trials. Often, children and parents in the third world countries do not know that they are receiving an experimental vaccine.  
Here is an example of clinical trials of Daptacel vaccine. To take part in the trial a child should be absolutely healthy, born after the 37th week, not sensitive to any of the vaccine components, without developmental delays, the family should not have a history of immune diseases, and so on. More or less the same requirements are stated for all other vaccine trials. That is, in comparison to medicines, which are usually tested on ill patients and are then given to ill patient, vaccines are tested exclusively on healthy children and are then given to both, healthy and not so healthy children, and even very sick children.
Safety and immunogenicity of a pentavalent vaccine in the US infants and toddlers and persistence of antibodies before a preschool booster dose: a randomized, clinical trial.
This is an article that reports the results of the aforementioned study:
Safety was tested for 30 and 60 days after each dose.
In 5.2% children from the test group and also in 5.2% from the control group (which received 3 different inoculations) serious adverse effects were noticed. Scientists decided that these adverse effects were not related to the inoculation. Authors did not report what adverse effects there were and why they were excluded.
Safety and immunogenicity of a modified process hepatitis B vaccine in healthy infants.
Pediatr Infect Dis J
Clinical trials of the Recombivax-HB vaccine against Hepatitis B. Safety was tested for 14 days.
77% of children had adverse events. 28 children (1.6%) had serious adverse events. One child died (SIDS). Authors report that probably his death was not related to the vaccination.
Safety and immunogenicity of a bivalent Haemophilus influenzae type b/ hepatitis B vaccine in healthy infants. Hib-HB Vaccine Study Group.
Pediatr Infect Dis J
Clinical trials of Comvax vaccine, against haemophilus influenzae and hepatitis B. Safety was tested for 14 days.
Serious adverse events occurred in 17 babies (1.9%). 3 children died (SIDS). Scientists concluded that all serious adverse events were not related to the vaccine.
Safety, reactogenicity and immunogenicity of the combined hexavalent tetanus, diphtheria, acellular pertussis, hepatitis B, inactivated poliovirus vaccine and Haemophilus influenzae type b conjugate vaccine, for primary immunization of infants.
Clinical trials of Infanrix Hexa vaccine. Safery was tested during 30 days.
Serious adverse events occurred in 79 infants (2.7%). Nearly all of them were not related to the vaccine. One infant died (SIDS). It was not related to the vaccine.
Randomized, Controlled, Multicenter Study of the Immunogenicity and Safety of a Fully Fluid Liquid Combination Diphtheria-Tetanus Toxoid-Five-Component Acellular Pertussis (DTaP5), Inactivated Poliovirus (IPV), and Haemophilus influenzae Type b (Hib) Vaccine Compared with a DTaP3-IPV/ Hib Vaccine Administered at 3, 5, and 12 Months of Age.
Clin Vaccine Immunol.
Clinical trials of pentavalent vaccines: Safety was tested for 30 days. 8.5% of infants had serious adverse events, almost all of them were not related to the vaccine.
Immunogenicity, Safety, and Tolerability of a Hexavalent Vaccine in Infants.
Clinical trials of hexavalent vaccine. Safety was tested for 6 months.
84 infants (5.9%) had adverse events. Two died. No relation to the vaccine.
This is, more or less, how majority of safety trials look. They are rarely last longer than few days or weeks. Most of the time they have serious adverse events occurring in absolutely healthy children during this short period of time, and they are rarely happen to be related to vaccination. They are not related because in the control group, which receives a different vaccine or the same vaccines without an antigen, same events are registered. In these short clinical trials it is impossible to find autoimmune, oncological or neurological diseases, as well as many other diseases, which are not rare results of vaccination (it will be proven later), but that are not possible to be diagnosed earlier than few months or few years after vaccination.
Moreover, vaccine inserts always state that there no trials of carcinogenicity of the medicine, mutagenecity and its possible effect on reproductive system were conducted.
By the way, in order for a child to be able to participate in clinical trials, he needs to be fully vaccinated with other vaccines. This is done in order to weed out very sensitive children, which have a tendency to become autistic, disabled or generally unhealthy after vaccination.
A few more examples
Full-text of papers mentioned above on Google.Drive